This is e-TeX, Version 3.141592-2.1 (MiKTeX 2.4) (preloaded format=latex 2004.3.15) 15 MAR 2004 14:53 entering extended mode **chapter1.tex (chapter1.tex LaTeX2e <2001/06/01> Babel and hyphenation patterns for english, french, german, ngerman, du mylang, nohyphenation, loaded. ! Undefined control sequence. l.1 \chapter {Mordell-Weil Theorem and Principal Homogeneous Spaces} ? q OK, entering \batchmode... ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.1 \chapter{M ordell-Weil Theorem and Principal Homogeneous Spaces} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! 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Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no H in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no 0 in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 5--17 [][] [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 5--17 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (19.9332pt too wide) in paragraph at lines 5--17 \OT1/cmr/m/it/10 open [] Overfull \hbox (22.10538pt too wide) in paragraph at lines 5--17 \OT1/cmr/m/it/10 prob- [] Overfull \hbox (15.33324pt too wide) in paragraph at lines 5--17 \OT1/cmr/m/it/10 lem [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.18 I n section 1, we introduce the Kummer pairing and prove its main You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no K in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! ! Undefined control sequence. l.21 ...gy. In section 2, we study in details\edit {detail} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! ! Undefined control sequence. \Gk l.23 ...te index subgroup of the Galois group $\Gk $. Several The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no D in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 18--27 [][] [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 18--27 \OML/cmm/m/it/10 S$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.28 S ection 3 is devoted to the classical techniques for computing the You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no S in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no 3 in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! ! Undefined control sequence. l.29 weak Mordell-Weil\edit {group} by constructing a pairing $b : E(K) / mE(K) The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! ! Undefined control sequence. l.30 \times E[m] \ra K^* / {K^*}^m$ out of the Weil pairing and using\edit{an} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! ! Undefined control sequence. l.30 ...^m$ out of the Weil pairing and using\edit {an} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no 4 in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 28--34 [][] [] Overfull \hbox (9.84715pt too wide) in paragraph at lines 28--34 \OML/cmm/m/it/10 b \OT1/cmr/m/n/10 : [] Overfull \hbox (71.44684pt too wide) in paragraph at lines 28--34 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )\OML/cmm/ m/it/10 =mE\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/1 0  [] Overfull \hbox (62.50127pt too wide) in paragraph at lines 28--34 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]\OML/cmm/ m/it/10 K[]=[][]$ [] Overfull \hbox (19.33565pt too wide) in paragraph at lines 28--34 \OML/cmm/m/it/10 m \OT1/cmr/m/n/10 = [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 28--34 \OT1/cmr/m/n/10 2$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.35 I n section 5, we define Selmer group\edit{groups} and Shafarevich-Tate ... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no 5 in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! ! Undefined control sequence. l.35 In section 5, we define Selmer group\edit {groups} and Shafarevich-Tate ... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! ! Undefined control sequence. l.35 ...it{groups} and Shafarevich-Tate group\edit {groups} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no - in font nullfont! ! Undefined control sequence. l.38 ...big open problems in number theory - \edit {should be an The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no : in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! ! Undefined control sequence. \Sha l.40 conjecture about the finiteness of $\Sha $. In section 6, we prove The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no 6 in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 35--44 [][] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.45 I n the last section, we introduce another method for computing the You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! ! Undefined control sequence. l.46 Selmer group which works in a\edit {delete word ``a''} greater generalit... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no J in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 45--48 [][] [] ! Undefined control sequence. l.49 \edit {Include a paragraph stating your assumptions about the background The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.49 \edit{I nclude a paragraph stating your assumptions about the background You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no Y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no Y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 49--54 [][] [] ! Undefined control sequence. l.58 \section {Weak Mordell-Weil Group and Kummer Pairing via The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.58 \section{W eak Mordell-Weil Group and Kummer Pairing via You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no K in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no P in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no y in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 58--60 [][] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.61 S uppose that $E / K$ is an elliptic curve over a number field and You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no S in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! ! Undefined control sequence. l.62 ...ger, such that $E[m] \subseteq E(k)$.\edit {What The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no W in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ? in font nullfont! Missing character: There is no Y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ? in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no ? in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! ! Undefined control sequence. l.69 ...l Mordell-Weil group $E(K)$. In fact,\edit {What?} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no W in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ? in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 61--70 [][] [] Overfull \hbox (22.16666pt too wide) in paragraph at lines 61--70 \OML/cmm/m/it/10 E=K$ [] Overfull \hbox (19.33565pt too wide) in paragraph at lines 61--70 \OML/cmm/m/it/10 m \OMS/cmsy/m/n/10  [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 61--70 \OT1/cmr/m/n/10 2$ [] Overfull \hbox (32.84953pt too wide) in paragraph at lines 61--70 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ] \OMS/cms y/m/n/10  [] Overfull \hbox (21.25696pt too wide) in paragraph at lines 61--70 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 k\OT1/cmr/m/n/10 )$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 61--70 \OML/cmm/m/it/10 k$ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 61--70 \OML/cmm/m/it/10 K$ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 61--70 \OML/cmm/m/it/10 K$ [] Overfull \hbox (22.16666pt too wide) in paragraph at lines 61--70 \OML/cmm/m/it/10 E=K$ [] Overfull \hbox (63.66904pt too wide) in paragraph at lines 61--70 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )\OML/cmm/ m/it/10 =mE\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 61--70 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (7.95831pt too wide) in paragraph at lines 61--70 \OML/cmm/m/it/10 E$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 61--70 \OML/cmm/m/it/10 m$ [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 61--70 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.72 D efine a pairing You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no D in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 72--73 [][] [] ! Undefined control sequence. l.74 ... \times \textrm{Gal}(\overline{K} / K) \ra E[m], The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Overfull \hbox (118.70834pt too wide) detected at line 75 \OML/cmm/m/it/10  \OT1/cmr/m/n/10 : \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cm m/m/it/10 K\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10  []\OT1/cmr/m/n/10 ([]\OML/cmm/m /it/10 =K\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m \OT1/cmr/m/n/10 ]\OML/cmm/m/it/10 ; [] Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no : in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! ! Undefined control sequence. l.76 ...or each $P \in E(K)$ choose $Q \in E$\edit { The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (17.25343pt too wide) in paragraph at lines 75--79 \OML/cmm/m/it/10 P \OMS/cmsy/m/n/10 2 [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 75--79 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (17.34995pt too wide) in paragraph at lines 75--79 \OML/cmm/m/it/10 Q \OMS/cmsy/m/n/10 2 [] Overfull \hbox (7.95831pt too wide) in paragraph at lines 75--79 \OML/cmm/m/it/10 E$ [] Overfull \hbox (17.34995pt too wide) in paragraph at lines 75--79 \OML/cmm/m/it/10 Q \OMS/cmsy/m/n/10 2 [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 75--79 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 ([])$ [] Overfull \hbox (7.95831pt too wide) in paragraph at lines 75--79 \OML/cmm/m/it/10 E$ [] Overfull \hbox (32.79677pt too wide) in paragraph at lines 75--79 \OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]\OML/cmm/m/it/10 Q \OT1/cmr /m/n/10 = [] Overfull \hbox (7.80904pt too wide) in paragraph at lines 75--79 \OML/cmm/m/it/10 P$ [] Overfull \hbox (44.0879pt too wide) in paragraph at lines 75--79 \OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 P; \OT1/cmr/m/n/10 ) := [] Overfull \hbox (21.06845pt too wide) in paragraph at lines 75--79 \OML/cmm/m/it/10 Q[] \OMS/cmsy/m/n/10 [] Overfull \hbox (7.90555pt too wide) in paragraph at lines 75--79 \OML/cmm/m/it/10 Q$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.80 F irst of all, this pairing is well-defined. Indeed, suppose that You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no F in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! 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Missing character: There is no c in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no H in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no K in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 80--87 [][] [] Overfull \hbox (10.7111pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q[]$ [] Overfull \hbox (32.79677pt too wide) in paragraph at lines 80--87 \OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]\OML/cmm/m/it/10 Q \OT1/cmr /m/n/10 = [] Overfull \hbox (35.60233pt too wide) in paragraph at lines 80--87 \OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]\OML/cmm/m/it/10 Q[] \OT1/c mr/m/n/10 = [] Overfull \hbox (7.80904pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 P$ [] Overfull \hbox (23.37401pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q[] \OMS/cmsy/m/n/10 [] Overfull \hbox (21.26662pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q[] \OT1/cmr/m/n/10 = [] Overfull \hbox (21.06845pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q[] \OMS/cmsy/m/n/10 [] Overfull \hbox (7.90555pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q$ [] Overfull \hbox (36.71352pt too wide) in paragraph at lines 80--87 \OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ](\OML/cmm/m/it/10 Q[] \OMS/ cmsy/m/n/10 [] Overfull \hbox (22.34996pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q\OT1/cmr/m/n/10 ) = [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 80--87 \OT1/cmr/m/n/10 0$ [] Overfull \hbox (15.68335pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q \OMS/cmsy/m/n/10 [] Overfull \hbox (20.1555pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q[] \OMS/cmsy/m/n/10 2 [] Overfull \hbox (32.84953pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ] \OMS/cms y/m/n/10  [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (18.4889pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q[] \OMS/cmsy/m/n/10 [] Overfull \hbox (7.90555pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q$ [] Overfull \hbox (31.19447pt too wide) in paragraph at lines 80--87 \OT1/cmr/m/n/10 ([]\OML/cmm/m/it/10 =K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (22.3778pt too wide) in paragraph at lines 80--87 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 Q[] \OMS/cmsy/m/n/10 [] Overfull \hbox (27.73506pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q\OT1/cmr/m/n/10 )[] = [] Overfull \hbox (18.4889pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q[] \OMS/cmsy/m/n/10 [] Overfull \hbox (7.90555pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q$ [] Overfull \hbox (23.37401pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q[] \OMS/cmsy/m/n/10 [] Overfull \hbox (21.26662pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q[] \OT1/cmr/m/n/10 = [] Overfull \hbox (21.06845pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q[] \OMS/cmsy/m/n/10 [] Overfull \hbox (7.90555pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 Q$ [] Overfull \hbox (5.76158pt too wide) in paragraph at lines 80--87 \OML/cmm/m/it/10 $ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.88 T he basic properties of $\kappa$ are summarized in the following You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no : in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 88--90 [][] [] Overfull \hbox (5.76158pt too wide) in paragraph at lines 88--90 \OML/cmm/m/it/10 $ [] ! LaTeX Error: Environment prop undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.91 \begin{prop} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.92 T he pairing $\kappa$ is bilinear, with left kernel equal to You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! ! Undefined control sequence. l.97 bilinear pairing\edit {The Gal below should be in roman!} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no ! in font nullfont! 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Undefined control sequence. l.99 E(K) / mE(K) \times \textrm{Gal}(L / K) \ra E[m]. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Overfull \hbox (140.9354pt too wide) detected at line 100 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )\OML/cmm/ m/it/10 =mE\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/1 0  []\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 L=K\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 E\ OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]\OML/cmm/m/it/10 : [] ! LaTeX Error: \begin{document} ended by \end{prop}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.101 \end{prop} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.103 \verb "Proof:" Bilinearity of $\kappa$ is obvious from the You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no B in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! 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Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.111 L et $\sigma \in \Gk(K)$ be in the right kernel. In this case it You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! ! Undefined control sequence. l.111 Let $\sigma \in \Gk (K)$ be in the right kernel. In this case it The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! 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Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! 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Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! 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Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 111--119 [][] [] Overfull \hbox (15.5173pt too wide) in paragraph at lines 111--119 \OML/cmm/m/it/10  \OMS/cmsy/m/n/10 2 [] Overfull \hbox (16.98613pt too wide) in paragraph at lines 111--119 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (6.0729pt too wide) in paragraph at lines 111--119 \OML/cmm/m/it/10 $ [] Overfull \hbox (21.01392pt too wide) in paragraph at lines 111--119 \OML/cmm/m/it/10 L=K$ [] Overfull \hbox (17.25343pt too wide) in paragraph at lines 111--119 \OML/cmm/m/it/10 P \OMS/cmsy/m/n/10 2 [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 111--119 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (7.90555pt too wide) in paragraph at lines 111--119 \OML/cmm/m/it/10 Q$ [] Overfull \hbox (32.79677pt too wide) in paragraph at lines 111--119 \OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]\OML/cmm/m/it/10 Q \OT1/cmr /m/n/10 = [] Overfull \hbox (7.80904pt too wide) in paragraph at lines 111--119 \OML/cmm/m/it/10 P$ [] Overfull \hbox (41.31012pt too wide) in paragraph at lines 111--119 \OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 P; \OT1/cmr/m/n/10 ) = [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 111--119 \OT1/cmr/m/n/10 0$ [] Overfull \hbox (23.84616pt too wide) in paragraph at lines 111--119 \OML/cmm/m/it/10 Q[] \OT1/cmr/m/n/10 = [] Overfull \hbox (7.90555pt too wide) in paragraph at lines 111--119 \OML/cmm/m/it/10 Q$ [] Overfull \hbox (50.0163pt too wide) in paragraph at lines 111--119 \OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ][]\OML/cmm/m/it/10 E\OT1/cm r/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (6.80557pt too wide) in paragraph at lines 111--119 \OML/cmm/m/it/10 L$ [] Overfull \hbox (6.0729pt too wide) in paragraph at lines 111--119 \OML/cmm/m/it/10 $ [] Overfull \hbox (15.5173pt too wide) in paragraph at lines 111--119 \OML/cmm/m/it/10  \OMS/cmsy/m/n/10 2 [] Overfull \hbox (44.41675pt too wide) in paragraph at lines 111--119 []\OT1/cmr/m/n/10 ([]\OML/cmm/m/it/10 =L\OT1/cmr/m/n/10 )$ [] Overfull \hbox (15.5173pt too wide) in paragraph at lines 111--119 \OML/cmm/m/it/10  \OMS/cmsy/m/n/10 2 [] Overfull \hbox (44.41675pt too wide) in paragraph at lines 111--119 []\OT1/cmr/m/n/10 ([]\OML/cmm/m/it/10 =L\OT1/cmr/m/n/10 )$ [] Overfull \hbox (50.0163pt too wide) in paragraph at lines 111--119 \OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ][]\OML/cmm/m/it/10 E\OT1/cm r/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.120 W e obtain the perfect bilinear pairing by moding out by the left You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.121 and right kernels of $\kappa$. $\hfill \Box $ \\ Load the latexsym or the amsfonts package to define this symbol Overfull \hbox (20.0pt too wide) in paragraph at lines 120--122 [][] [] Overfull \hbox (5.76158pt too wide) in paragraph at lines 120--122 \OML/cmm/m/it/10 $ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.123 N ext, our goal is to describe the Kummer pairing in terms of You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no N in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no K in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 123--127 [][] [] Overfull \hbox (31.19447pt too wide) in paragraph at lines 123--127 \OT1/cmr/m/n/10 ([]\OML/cmm/m/it/10 =K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (19.33565pt too wide) in paragraph at lines 123--127 \OML/cmm/m/it/10 m > [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 123--127 \OT1/cmr/m/n/10 1$ [] ! LaTeX Error: The font size command \normalsize is not defined: there is probably something wrong with the class file. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.127 \[ Your command was ignored. Type I to replace it with another command, or to continue without it. [1] ! LaTeX Error: Environment CD undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.128 \begin{CD} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Undefined control sequence. l.129 0 \ra E[m] @>>> E(\overline{K}) @>[m]>> E(\overline{K}) \ra 0. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.129 ...\overline{K}) @>[m]>> E(\overline{K}) \ra 0. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: \begin{document} ended by \end{CD}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.130 \end{CD} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (178.1851pt too wide) detected at line 131 \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/ m/n/10 ]@ \OML/cmm/m/it/10 >>> E\OT1/cmr/m/n/10 ([])@ \OML/cmm/m/it/10 > \OT1/c mr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ] \OML/cmm/m/it/10 >> E\OT1/cmr/m/ n/10 ([])0\OML/cmm/m/it/10 : [] Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no y in font nullfont! ! LaTeX Error: Environment CD undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.135 \begin{CD} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Undefined control sequence. l.136 0 @>>> H^0(\Gk , E[m]) @>>> H^0(\Gk, E(\overline{K})) @>[m]>> The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.136 0 @>>> H^0(\Gk, E[m]) @>>> H^0(\Gk , E(\overline{K})) @>[m]>> The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.137 H^0(\Gk , E(\overline{K})) \\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.138 @>\delta>> H^1(\Gk , E[m]) @>>> H^1(\Gk, E(\overline{K})) @>[m]>> The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.138 @>\delta>> H^1(\Gk, E[m]) @>>> H^1(\Gk , E(\overline{K})) @>[m]>> The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.139 H^1(\Gk , E(\overline{K})). The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: \begin{document} ended by \end{CD}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.140 \end{CD} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (577.30383pt too wide) detected at line 141 \OT1/cmr/m/n/10 0@ \OML/cmm/m/it/10 >>> H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ])@ \OML/cmm/m/it/10 >>> H[ ]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 ([]))@ \OML/cmm/m/it/10 > \OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ] \OML/cmm/m/it/10 >> H[]\O T1/cmr/m/n/10 (\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 ([]))[] @ \OML/cmm/m/it/10 >  >> H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ])@ \OML/cmm/m/it/10 >>> H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 ([]))@ \OML/cmm/m/it/10 > \OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ] \OML/cmm/m/it/10 >> H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 ([]))\OML/cmm/m/it/10 : [] Missing character: There is no B in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Overfull \hbox (55.04297pt too wide) in paragraph at lines 141--144 \OML/cmm/m/it/10 H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 G; M\OT1/cmr/m/n/10 ) = [] Overfull \hbox (17.51764pt too wide) in paragraph at lines 141--144 \OML/cmm/m/it/10 M[]$ [] Overfull \hbox (7.86249pt too wide) in paragraph at lines 141--144 \OML/cmm/m/it/10 G$ [] Overfull \hbox (7.86249pt too wide) in paragraph at lines 141--144 \OML/cmm/m/it/10 G$ [] Overfull \hbox (10.79166pt too wide) in paragraph at lines 141--144 \OML/cmm/m/it/10 M$ [] ! LaTeX Error: Environment CD undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.145 \begin{CD} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Undefined control sequence. l.148 @>\delta>> H^1(\Gk , E[m]) @>>> H^1(\Gk, E(\overline{K})) @>[m]>> The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.148 @>\delta>> H^1(\Gk, E[m]) @>>> H^1(\Gk , E(\overline{K})) @>[m]>> The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.149 H^1(\Gk , E(\overline{K})). The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: \begin{document} ended by \end{CD}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.150 \end{CD} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (516.79002pt too wide) detected at line 151 \OT1/cmr/m/n/10 0@ \OML/cmm/m/it/10 >>> E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT 1/cmr/m/n/10 )[\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]@ \OML/cmm/m/it/10 >>> E\OT1/ cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )@ \OML/cmm/m/it/10 > \OT1/cmr/m/ n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ] \OML/cmm/m/it/10 >> E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )[] @ \OML/cmm/m/it/10 >  >> H[]\OT1/cmr/m/ n/10 (\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]) @ \OML/cmm/m/it/10 >>> H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 ([]))@ \OML/cmm/m/it/10 > \OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ] \OML/cmm/m/it/10 >> H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 ([] ))\OML/cmm/m/it/10 : [] Missing character: There is no N in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no : in font nullfont! Overfull \hbox (15.37842pt too wide) in paragraph at lines 151--155 \OML/cmm/m/it/10  \OT1/cmr/m/n/10 = [] Overfull \hbox (33.72458pt too wide) in paragraph at lines 151--155 \OML/cmm/m/it/10 mE\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (24.81096pt too wide) in paragraph at lines 151--155 \OT1/cmr/m/it/10 Kum- [] Overfull \hbox (16.99432pt too wide) in paragraph at lines 151--155 \OT1/cmr/m/it/10 mer [] Overfull \hbox (12.26657pt too wide) in paragraph at lines 151--155 \OT1/cmr/m/it/10 se- [] Overfull \hbox (29.62894pt too wide) in paragraph at lines 151--155 \OT1/cmr/m/it/10 quence [] ! LaTeX Error: Environment CD undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.156 \begin{CD} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Undefined control sequence. l.157 0 \ra E(K) / mE(K) @>\delta>> H^1(\Gk, E[m]) @>>> H^1(\Gk, The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.157 0 \ra E(K) / mE(K) @>\delta>> H^1(\Gk , E[m]) @>>> H^1(\Gk, The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.157 ...) @>\delta>> H^1(\Gk, E[m]) @>>> H^1(\Gk , The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.158 E(\overline{K}))[m] \ra 0. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: \begin{document} ended by \end{CD}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.159 \end{CD} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (273.39923pt too wide) detected at line 160 \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/ m/n/10 )\OML/cmm/m/it/10 =mE\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )@ \OML/cmm/m/it/10 >  >> H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ; E\OT1/cmr/m/n /10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ])@ \OML/cmm/m/it/10 >>> H[]\OT1/cmr/m/n /10 (\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 ([]))[\OML/cmm/m/it/10 m\OT1/cmr/m/n/1 0 ]0\OML/cmm/m/it/10 : [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.162 S ince the left kernel of the pairing $\kappa$ is $mE(K)$, then You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 162--164 [][] [] Overfull \hbox (5.76158pt too wide) in paragraph at lines 162--164 \OML/cmm/m/it/10 $ [] Overfull \hbox (33.72458pt too wide) in paragraph at lines 162--164 \OML/cmm/m/it/10 mE\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (5.76158pt too wide) in paragraph at lines 162--164 \OML/cmm/m/it/10 $ [] ! Undefined control sequence. l.165 \delta_E : E(K) / mE(K) \ra \textrm{Hom}(\Gk, E[m]). The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.165 \delta_E : E(K) / mE(K) \ra \textrm{Hom}(\Gk , E[m]). The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Overfull \hbox (141.32967pt too wide) detected at line 166 \OML/cmm/m/it/10 [] \OT1/cmr/m/n/10 : \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/ cmm/m/it/10 K\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 =mE\OT1/cmr/m/n/10 (\OML/cmm/m/i t/10 K\OT1/cmr/m/n/10 )[](\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ])\OML/cmm/m/it/10 : [] Missing character: There is no U in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (11.20001pt too wide) in paragraph at lines 166--170 \OML/cmm/m/it/10 []$ [] Overfull \hbox (4.8229pt too wide) in paragraph at lines 166--170 \OML/cmm/m/it/10 $ [] ! Undefined control sequence. l.173 \section {Properties of the Field Extension $L = K([m]^{-1}E(K))/K$} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.173 \section{P roperties of the Field Extension $L = K([m]^{-1}E(K))/K$} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no P in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 173--174 [][] [] Overfull \hbox (17.36108pt too wide) in paragraph at lines 173--174 \OML/cmm/m/it/10 L \OT1/cmr/m/n/10 = [] Overfull \hbox (81.21077pt too wide) in paragraph at lines 173--174 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 ([\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ][]\OML/c mm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 ))\OML/cmm/m/it/ 10 =K$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.175 A fter introducing the Kummer pairing in the previous section, we will You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no A in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no K in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! ! Undefined control sequence. l.181 ... that $L / K$ is a finite extension.\edit {You might want to make The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no Y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ! in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no H in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no ! in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 175--191 [][] [] Overfull \hbox (17.36108pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 L \OT1/cmr/m/n/10 = [] Overfull \hbox (67.00243pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 ([\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ][]\OML/c mm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 ))$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 m$ [] Overfull \hbox (5.57639pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 $ [] Overfull \hbox (21.01392pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 L=K$ [] Overfull \hbox (17.36108pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 L \OT1/cmr/m/n/10 = [] Overfull \hbox (67.00243pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 ([\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ][]\OML/c mm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 ))$ [] Overfull \hbox (6.80557pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 L$ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 K$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 m$ [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (6.80557pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 L$ [] Overfull \hbox (6.80557pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 L$ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 K$ [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (6.80557pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 L$ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 K$ [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 175--191 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.192 T he main properties of the field extension $L / K$ are summarized You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 192--194 [][] [] Overfull \hbox (21.01392pt too wide) in paragraph at lines 192--194 \OML/cmm/m/it/10 L=K$ [] ! LaTeX Error: Environment prop undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.195 \begin{prop} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.196 ( i) The field extension $L / K$ is an abelian extension of You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no ( in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! ! Undefined control sequence. l.197 ...$. In other words, the Galois group \edit {Gal should The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no . in font nullfont! ! Undefined control sequence. l.199 ...very element has order dividing $m$.\edit {This The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no Y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! ! Undefined control sequence. l.202 ... If $S$ is the finite set of places,\edit {Remove The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no R in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{prop}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.207 \end{prop} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 196--208 [][] [] Overfull \hbox (21.01392pt too wide) in paragraph at lines 196--208 \OML/cmm/m/it/10 L=K$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 196--208 \OML/cmm/m/it/10 m$ [] Overfull \hbox (28.79172pt too wide) in paragraph at lines 196--208 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 L=K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 196--208 \OML/cmm/m/it/10 m$ [] Overfull \hbox (17.36108pt too wide) in paragraph at lines 196--208 \OML/cmm/m/it/10 L \OT1/cmr/m/n/10 = [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 196--208 \OML/cmm/m/it/10 K$ [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 196--208 \OT1/cmr/m/n/10 1$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 196--208 \OML/cmm/m/it/10 m$ [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 196--208 \OML/cmm/m/it/10 S$ [] Overfull \hbox (7.95831pt too wide) in paragraph at lines 196--208 \OML/cmm/m/it/10 E$ [] Overfull \hbox (5.57639pt too wide) in paragraph at lines 196--208 \OML/cmm/m/it/10 $ [] Overfull \hbox (32.68983pt too wide) in paragraph at lines 196--208 \OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ) \OMS/cms y/m/n/10 6\OT1/cmr/m/n/10 = [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 196--208 \OT1/cmr/m/n/10 0$ [] Overfull \hbox (21.01392pt too wide) in paragraph at lines 196--208 \OML/cmm/m/it/10 L=K$ [] Overfull \hbox (15.02078pt too wide) in paragraph at lines 196--208 \OML/cmm/m/it/10  [] [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 196--208 \OML/cmm/m/it/10 S$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.209 T he following lemma will be used in the proof of the proposition: You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no : in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 209--210 [][] [] ! LaTeX Error: Environment lem undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.211 \begin{lem} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.212 S uppose that $\nu$ is a discrete valuation, such that $\nu(m) = 0$ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no S in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! ! Undefined control sequence. l.213 and $E / K$ has a\edit {Delete ``a''} good reduction at $\nu$. Then the... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no D in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! ! Undefined control sequence. l.214 $E(K)[m] \ra \tilde{E}_\nu(k_\nu)$ is injective. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{lem}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.215 \end{lem} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 212--216 [][] [] Overfull \hbox (5.57639pt too wide) in paragraph at lines 212--216 \OML/cmm/m/it/10 $ [] Overfull \hbox (32.68983pt too wide) in paragraph at lines 212--216 \OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ) = [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 212--216 \OT1/cmr/m/n/10 0$ [] Overfull \hbox (22.16666pt too wide) in paragraph at lines 212--216 \OML/cmm/m/it/10 E=K$ [] Overfull \hbox (5.57639pt too wide) in paragraph at lines 212--216 \OML/cmm/m/it/10 $ [] Overfull \hbox (69.59738pt too wide) in paragraph at lines 212--216 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )[\OML/cmm /m/it/10 m\OT1/cmr/m/n/10 ][](\OML/cmm/m/it/10 k[]\OT1/cmr/m/n/10 )$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.217 \verb "Proof:" This is proved in [Sil-1, VIII.\S1]. $\hfill \Box$ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no [ in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no V in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no . in font nullfont! LaTeX Font Info: Try loading font information for OMS+cmr on input line 217. (C:\texmf\tex\latex\base\omscmr.fd File: omscmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions ) LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <10> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 217. Missing character: There is no 1 in font nullfont! Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.217 ...proved in [Sil-1, VIII.\S1]. $\hfill \Box $ Load the latexsym or the amsfonts package to define this symbol Overfull \hbox (51.49973pt too wide) in paragraph at lines 217--219 [][][]\OT1/cmtt/m/n/10 Proof: [] Overfull \hbox (4.44447pt too wide) in paragraph at lines 217--219 \OMS/cmsy/m/n/10 x [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.220 W e are now ready to prove the proposition: You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no : in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 220--221 [][] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.222 \verb "Proof of proposition 2.1:" (i) This is a consequence of You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no ( in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! ! Undefined control sequence. l.224 \cdot)$ is an injection Gal$(L / K) \ra \textrm{Hom}(E(K), E[m])$. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.243 proves the proposition. $\hfill \Box $ \\ Load the latexsym or the amsfonts package to define this symbol Overfull \hbox (151.24886pt too wide) in paragraph at lines 222--244 [][][]\OT1/cmtt/m/n/10 Proof of proposition 2.1: [] Overfull \hbox (18.85063pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10  \OMS/cmsy/m/n/10 7! [] Overfull \hbox (26.27893pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 \OT1/c mr/m/n/10 )$ [] Overfull \hbox (109.08578pt too wide) in paragraph at lines 222--244 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 L=K\OT1/cmr/m/n/10 )[](\OML/cmm/m/it/10 E\OT1 /cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; E\OT1/cmr/m/ n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ])$ [] Overfull \hbox (28.79172pt too wide) in paragraph at lines 222--244 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 L=K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 m$ [] Overfull \hbox (80.29407pt too wide) in paragraph at lines 222--244 []\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cm r/m/n/10 )\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/1 0 ])$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 m$ [] Overfull \hbox (17.34995pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 Q \OMS/cmsy/m/n/10 2 [] Overfull \hbox (50.0163pt too wide) in paragraph at lines 222--244 \OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ][]\OML/cmm/m/it/10 E\OT1/cm r/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (18.36455pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 P \OT1/cmr/m/n/10 = [] Overfull \hbox (22.24126pt too wide) in paragraph at lines 222--244 \OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]\OML/cmm/m/it/10 Q$ [] Overfull \hbox (17.36108pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 L \OT1/cmr/m/n/10 = [] Overfull \hbox (24.89168pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 Q\OT1/cmr/m/n/10 )$ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 K$ [] Overfull \hbox (15.02078pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10  [] [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 S$ [] Overfull \hbox (8.38194pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 []$ [] Overfull \hbox (5.57639pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 $ [] Overfull \hbox (24.89168pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 Q\OT1/cmr/m/n/10 )$ [] Overfull \hbox (24.53271pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 D[]$ [] Overfull \hbox (20.6494pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 I[]$ [] Overfull \hbox (20.6494pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 I[]$ [] Overfull \hbox (24.89168pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 Q\OT1/cmr/m/n/10 )$ [] Overfull \hbox (20.6494pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 I[]$ [] Overfull \hbox (33.02211pt too wide) in paragraph at lines 222--244 []\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 k[]\OT1/cmr/m/n/10 )$ [] Overfull \hbox (12.88664pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 k[]$ [] Overfull \hbox (24.89168pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 Q\OT1/cmr/m/n/10 )$ [] Overfull \hbox (8.38194pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 []$ [] Overfull \hbox (24.95735pt too wide) in paragraph at lines 222--244 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 Q[] \OMS/cmsy/m/n/10 [] Overfull \hbox (29.09998pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 Q\OT1/cmr/m/n/10 )[] = [] Overfull \hbox (21.06845pt too wide) in paragraph at lines 222--244 [] \OMS/cmsy/m/n/10 [] Overfull \hbox (18.46106pt too wide) in paragraph at lines 222--244 [] \OT1/cmr/m/n/10 = [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 222--244 []$ [] Overfull \hbox (15.5173pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10  \OMS/cmsy/m/n/10 2 [] Overfull \hbox (20.6494pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 I[]$ [] Overfull \hbox (21.06845pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 Q[] \OMS/cmsy/m/n/10 [] Overfull \hbox (17.34995pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 Q \OMS/cmsy/m/n/10 2 [] Overfull \hbox (22.29402pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]$ [] Overfull \hbox (17.34995pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 Q \OMS/cmsy/m/n/10 2 [] Overfull \hbox (50.0163pt too wide) in paragraph at lines 222--244 \OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ][]\OML/cmm/m/it/10 E\OT1/cm r/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (23.84616pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 Q[] \OT1/cmr/m/n/10 = [] Overfull \hbox (7.90555pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 Q$ [] Overfull \hbox (20.6494pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 I[]$ [] Overfull \hbox (39.10002pt too wide) in paragraph at lines 222--244 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 Q\OT1/cmr/m/n/10 )\OML/cmm/ m/it/10 =K$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.245 O ur goal in this section is to show that $L / K$ is a finite You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no O in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! ! Undefined control sequence. l.246 ...So far, we concluded that $L / K$ is\edit {an} abelian The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. l.249 claim\edit {deduce} the finiteness of $L/K$. The next theorem The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 245--252 [][] [] Overfull \hbox (21.01392pt too wide) in paragraph at lines 245--252 \OML/cmm/m/it/10 L=K$ [] Overfull \hbox (21.01392pt too wide) in paragraph at lines 245--252 \OML/cmm/m/it/10 L=K$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 245--252 \OML/cmm/m/it/10 m$ [] Overfull \hbox (21.01392pt too wide) in paragraph at lines 245--252 \OML/cmm/m/it/10 L=K$ [] ! LaTeX Error: Environment thm undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.253 \begin{thm} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.254 L et $K$ be a number field, $m \geq 2$ be an integer, and $S$ -\edit{Why You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no - in font nullfont! ! Undefined control sequence. l.254 ... $m \geq 2$ be an integer, and $S$ -\edit {Why The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no W in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ? in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{thm}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.261 \end{thm} Your command was ignored. Type I to replace it with another command, or to continue without it. 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.263 \verb "Proof:" If the proposition is true for a finite extension You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! 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Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.272 W e define \emph{the ring of $S$-integers} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! 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Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no P in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no D in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (6.70831pt too wide) in paragraph at lines 275--280 \OML/cmm/m/it/10 S$ [] Overfull \hbox (13.38597pt too wide) in paragraph at lines 275--280 \OML/cmm/m/it/10 R[]$ [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 275--280 \OML/cmm/m/it/10 S$ [] Overfull \hbox (6.80557pt too wide) in paragraph at lines 275--280 \OML/cmm/m/it/10 L$ [] Overfull \hbox (13.38597pt too wide) in paragraph at lines 275--280 \OML/cmm/m/it/10 R[]$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.281 N ext, we use another auxiliary result: You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no N in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no : in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 281--282 [][] [] ! LaTeX Error: Environment lem undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.283 \begin{lem} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.284 L et $K$ be a number field (more generally, any field of You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no 0 in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! ! Undefined control sequence. l.289 extension of $K$ of degree $m$.\edit {You mean ``of exponent~$m$.} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no Y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{lem}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.290 \end{lem} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 284--291 [][] [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 284--291 \OML/cmm/m/it/10 K$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 284--291 \OML/cmm/m/it/10 m$ [] Overfull \hbox (13.6216pt too wide) in paragraph at lines 284--291 \OML/cmm/m/it/10 []$ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 284--291 \OML/cmm/m/it/10 K$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 284--291 \OML/cmm/m/it/10 m$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 284--291 \OML/cmm/m/it/10 m$ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 284--291 \OML/cmm/m/it/10 K$ [] Overfull \hbox (17.36108pt too wide) in paragraph at lines 284--291 \OML/cmm/m/it/10 L \OT1/cmr/m/n/10 = [] Overfull \hbox (39.61809pt too wide) in paragraph at lines 284--291 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 a[] \OT1/cmr/m/n/10 : [] Overfull \hbox (14.73029pt too wide) in paragraph at lines 284--291 \OML/cmm/m/it/10 a \OMS/cmsy/m/n/10 2 [] Overfull \hbox (13.09723pt too wide) in paragraph at lines 284--291 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 284--291 \OML/cmm/m/it/10 K$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 284--291 \OML/cmm/m/it/10 m$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 284--291 \OML/cmm/m/it/10 m$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.292 \verb "Proof:" *** LATER *** $\hfill \Box$ \\ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no * in font nullfont! Missing character: There is no * in font nullfont! Missing character: There is no * in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no * in font nullfont! Missing character: There is no * in font nullfont! Missing character: There is no * in font nullfont! ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.292 \verb"Proof:" *** LATER *** $\hfill \Box $ \\ Load the latexsym or the amsfonts package to define this symbol Overfull \hbox (51.49973pt too wide) in paragraph at lines 292--293 [][][]\OT1/cmtt/m/n/10 Proof: [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.294 A ccording to the lemma 2.4\edit{Change ``the lemma 2.4'' to ``Lemma 2.... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no A in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no 4 in font nullfont! ! Undefined control sequence. l.294 According to the lemma 2.4\edit {Change ``the lemma 2.4'' to ``Lemma 2.... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no C in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no 4 in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no 4 in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! ! Undefined control sequence. l.296 of $S$.\edit {Remind reader that $L$ is now exponent $m$ abelian and The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no R in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.301 S uppose that $\nu \notin S$. Then $a^{1/m} \in L$ for some $a \in You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no S in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! 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Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.311 W e will be done if we prove that $T_S$ is finite. The idea is to You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! ! Undefined control sequence. l.312 consider the natural map $R_S^* \ra T_S$. We claim that this map The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no H in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! ! Undefined control sequence. l.320 the map $R_S^* \ra T_S$ is surjective. Since its kernel contains The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! ! Undefined control sequence. l.321 ...n a surjective map $R_S^* / (R_S^*)^m \ra The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no D in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no [ in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no V in font nullfont! Missing character: There is no ] in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.325 finite extension. $\hfill \Box $ \\ Load the latexsym or the amsfonts package to define this symbol Overfull \hbox (20.0pt too wide) in paragraph at lines 311--326 [][] [] Overfull \hbox (11.63683pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 T[]$ [] Overfull \hbox (25.0228pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 R[]T[]$ [] Overfull \hbox (15.02078pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10  [] [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 S$ [] Overfull \hbox (13.38597pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 R[]$ [] Overfull \hbox (14.73029pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 a \OMS/cmsy/m/n/10 2 [] Overfull \hbox (13.80556pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 K[]$ [] Overfull \hbox (11.63683pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 T[]$ [] Overfull \hbox (18.67186pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 aR[]$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 m$ [] Overfull \hbox (13.38597pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 R[]$ [] Overfull \hbox (11.63683pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 T[]$ [] Overfull \hbox (13.38597pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 R[]$ [] Overfull \hbox (29.22737pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 aR[] \OT1/cmr/m/n/10 = [] Overfull \hbox (25.27374pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 b[]R[]$ [] Overfull \hbox (13.73605pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 b \OMS/cmsy/m/n/10 2 [] Overfull \hbox (13.80556pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 K[]$ [] Overfull \hbox (15.8414pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 a \OT1/cmr/m/n/10 = [] Overfull \hbox (17.61235pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 ub[]$ [] Overfull \hbox (15.16898pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 u \OMS/cmsy/m/n/10 2 [] Overfull \hbox (13.38597pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 R[]$ [] Overfull \hbox (5.28589pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 a$ [] Overfull \hbox (5.72458pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 u$ [] Overfull \hbox (11.63683pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 T[]$ [] Overfull \hbox (25.0228pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 R[]T[]$ [] Overfull \hbox (28.75989pt too wide) in paragraph at lines 311--326 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 R[]\OT1/cmr/m/n/10 )[]$ [] Overfull \hbox (58.7827pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 R[]=\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 R[]\OT1/cmr/m/n/10 )[]\O ML/cmm/m/it/10 T[]$ [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 S$ [] Overfull \hbox (13.38597pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 R[]$ [] Overfull \hbox (47.14587pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 R[]=\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 R[]\OT1/cmr/m/n/10 )[]$ [] Overfull \hbox (11.63683pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 T[]$ [] Overfull \hbox (21.01392pt too wide) in paragraph at lines 311--326 \OML/cmm/m/it/10 L=K$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.327 W e finally proved that $L / K$ is a finite extension, which is You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! ! Undefined control sequence. l.329 ...K) / mE(K) \times \textrm{Gal}(L / K) \ra E[m]$ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 327--331 [][] [] Overfull \hbox (21.01392pt too wide) in paragraph at lines 327--331 \OML/cmm/m/it/10 L=K$ [] Overfull \hbox (63.66904pt too wide) in paragraph at lines 327--331 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )\OML/cmm/ m/it/10 =mE\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (71.44684pt too wide) in paragraph at lines 327--331 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )\OML/cmm/ m/it/10 =mE\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/1 0  [] Overfull \hbox (66.71077pt too wide) in paragraph at lines 327--331 []\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 L=K\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 E\OT1/ cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]$ [] Overfull \hbox (5.76158pt too wide) in paragraph at lines 327--331 \OML/cmm/m/it/10 $ [] ! Undefined control sequence. l.334 \section {Computation of the Weak Mordell-Weil Group and The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.334 \section{C omputation of the Weak Mordell-Weil Group and You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no P in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no H in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 334--336 [][] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.337 R ecall that we assumed in the very beginning that $E[m] \subset You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no R in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no H in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no 9 in font nullfont! Missing character: There is no 0 in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. l.339 follows from Hilbert 90 Satz theorem\edit {Doesn't satz mean theorem, The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no D in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ? in font nullfont! Missing character: There is no J in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no H in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no 9 in font nullfont! Missing character: There is no 0 in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no [ in font nullfont! Missing character: There is no J in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no ] in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. \Gk l.341 $\Gk \ra \mu_m$ has the form $\sigma \mapsto \sigma(\beta) / The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.341 $\Gk \ra \mu_m$ has the form $\sigma \mapsto \sigma(\beta) / The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. l.343 ...isomorphism $\delta_K : K^* / {K^*}^m \ra The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.344 \textrm{Hom}(\Gk , \mu_m)$. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 337--345 [][] [] Overfull \hbox (32.84953pt too wide) in paragraph at lines 337--345 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ] \OMS/cms y/m/n/10  [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 337--345 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (24.17711pt too wide) in paragraph at lines 337--345 \OML/cmm/m/it/10 [] \OMS/cmsy/m/n/10  [] Overfull \hbox (13.80556pt too wide) in paragraph at lines 337--345 \OML/cmm/m/it/10 K[]$ [] Overfull \hbox (13.6216pt too wide) in paragraph at lines 337--345 \OML/cmm/m/it/10 []$ [] Overfull \hbox (18.85063pt too wide) in paragraph at lines 337--345 \OML/cmm/m/it/10  \OMS/cmsy/m/n/10 7! [] Overfull \hbox (31.21877pt too wide) in paragraph at lines 337--345 \OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 \OT1/cmr/m/n/10 )\OML/cmm/ m/it/10 = $ [] Overfull \hbox (15.62842pt too wide) in paragraph at lines 337--345 \OML/cmm/m/it/10 \OMS/cmsy/m/n/10 2 [] Overfull \hbox (13.80556pt too wide) in paragraph at lines 337--345 [][]$ [] Overfull \hbox (23.22453pt too wide) in paragraph at lines 337--345 \OML/cmm/m/it/10 [] \OMS/cmsy/m/n/10 2 [] Overfull \hbox (13.80556pt too wide) in paragraph at lines 337--345 \OML/cmm/m/it/10 K[]$ [] Overfull \hbox (17.7097pt too wide) in paragraph at lines 337--345 \OML/cmm/m/it/10 [] \OT1/cmr/m/n/10 : [] Overfull \hbox (86.88445pt too wide) in paragraph at lines 337--345 \OML/cmm/m/it/10 K[]=[][][]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ; []\OT1/cmr/m/n/ 10 )$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.346 T he main idea for the computation of the weak Mordell-Weil group You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 346--349 [][] [] Overfull \hbox (63.66904pt too wide) in paragraph at lines 346--349 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )\OML/cmm/ m/it/10 =mE\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (11.20001pt too wide) in paragraph at lines 346--349 \OML/cmm/m/it/10 []$ [] Overfull \hbox (12.1542pt too wide) in paragraph at lines 346--349 \OML/cmm/m/it/10 []$ [] ! Undefined control sequence. l.350 b : E(K)/mE(K) \times E[m] \ra K^* / {K^*}^m, The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Overfull \hbox (149.35075pt too wide) detected at line 351 \OML/cmm/m/it/10 b \OT1/cmr/m/n/10 : \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cm m/m/it/10 K\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 =mE\OT1/cmr/m/n/10 (\OML/cmm/m/it/ 10 K\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10  \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OM L/cmm/m/it/10 m\OT1/cmr/m/n/10 ]\OML/cmm/m/it/10 K[]=[][]; [] Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.354 F or the construction of this pairing, we use the Weil pairing $e_m You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no F in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! ! Undefined control sequence. l.355 : E[m] \times E[m] \ra \mu_m$, defined in [Sil1-III.\S8]. Define The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no , in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no [ in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no 8 in font nullfont! Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no D in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 354--356 [][] [] Overfull \hbox (17.80788pt too wide) in paragraph at lines 354--356 \OML/cmm/m/it/10 e[] \OT1/cmr/m/n/10 : [] Overfull \hbox (30.07182pt too wide) in paragraph at lines 354--356 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ] \OMS/cms y/m/n/10  [] Overfull \hbox (35.91562pt too wide) in paragraph at lines 354--356 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]\OML/cmm/ m/it/10 []$ [] Overfull \hbox (4.44447pt too wide) in paragraph at lines 354--356 \OMS/cmsy/m/n/10 x [] Overfull \hbox (140.28777pt too wide) detected at line 358 \OML/cmm/m/it/10 b\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 P; Q\OT1/cmr/m/n/10 ) = \OM L/cmm/m/it/10 []\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 (\OML/cmm /m/it/10 []\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 P\OT1/cmr/m/n/10 )(\OMS/cmsy/m/n/ 10 \OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; Q\OT1/cmr/m/n/10 ))\OML/cmm/m/it/10 : [] Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. l.362 the left, then \edit {insert: ``there would exist a~$P$ such that...''} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no : in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! ! Undefined control sequence. l.363 for all $Q \in E[m]$ and all $\sigma \in \Gk $ it will follow that The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (12.1542pt too wide) in paragraph at lines 358--367 \OML/cmm/m/it/10 []$ [] Overfull \hbox (12.1542pt too wide) in paragraph at lines 358--367 \OML/cmm/m/it/10 []$ [] Overfull \hbox (7.80904pt too wide) in paragraph at lines 358--367 \OML/cmm/m/it/10 P$ [] Overfull \hbox (17.34995pt too wide) in paragraph at lines 358--367 \OML/cmm/m/it/10 Q \OMS/cmsy/m/n/10 2 [] Overfull \hbox (22.29402pt too wide) in paragraph at lines 358--367 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]$ [] Overfull \hbox (15.5173pt too wide) in paragraph at lines 358--367 \OML/cmm/m/it/10  \OMS/cmsy/m/n/10 2$ [] Overfull \hbox (73.69026pt too wide) in paragraph at lines 358--367 \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cm m/m/it/10 P; \OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; Q\OT1/cmr/m/n/10 ) = [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 358--367 \OT1/cmr/m/n/10 1$ [] Overfull \hbox (41.31012pt too wide) in paragraph at lines 358--367 \OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 P; \OT1/cmr/m/n/10 ) = [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 358--367 \OT1/cmr/m/n/10 0$ [] Overfull \hbox (17.25343pt too wide) in paragraph at lines 358--367 \OML/cmm/m/it/10 P \OMS/cmsy/m/n/10 2 [] Overfull \hbox (33.72458pt too wide) in paragraph at lines 358--367 \OML/cmm/m/it/10 mE\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.368 T he pairing $b$ is easily computable. The next proposition You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no . in font nullfont! ! Undefined control sequence. l.369 ...how one can compute the pairing $b$.\edit {replace The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: Environment prop undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.372 \begin{prop} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! ! Undefined control sequence. l.373 ...$ be the finite set of places $\nu$,\edit {remove The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 368--377 [][] [] Overfull \hbox (4.29166pt too wide) in paragraph at lines 368--377 \OML/cmm/m/it/10 b$ [] Overfull \hbox (4.29166pt too wide) in paragraph at lines 368--377 \OML/cmm/m/it/10 b$ [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 368--377 \OML/cmm/m/it/10 S$ [] Overfull \hbox (5.57639pt too wide) in paragraph at lines 368--377 \OML/cmm/m/it/10 $ [] Overfull \hbox (7.95831pt too wide) in paragraph at lines 368--377 \OML/cmm/m/it/10 E$ [] Overfull \hbox (8.78014pt too wide) in paragraph at lines 368--377 \OML/cmm/m/it/10 m$ [] Overfull \hbox (4.29166pt too wide) in paragraph at lines 368--377 \OML/cmm/m/it/10 b$ [] Overfull \hbox (256.82262pt too wide) detected at line 380 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S; m\OT1/cmr/m/n/10 ) = \OM S/cmsy/m/n/10 f\OML/cmm/m/it/10 b \OMS/cmsy/m/n/10 2 \OML/cmm/m/it/10 K[]=[][] \OT1/cmr/m/n/10 : \OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b\OT1/cmr /m/n/10 ) \OMS/cmsy/m/n/10  \OT1/cmr/m/n/10 0([]\OML/cmm/m/it/10 m\OT1/cmr/m/n /10 )[]\OML/cmm/m/it/10  [] S\OMS/cmsy/m/n/10 g [] Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! ! Undefined control sequence. l.382 functions, satisfying\edit {What is $F_Q$? Is it $f_Q$?} div$(F_Q) = ... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no W in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ? in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ? in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{prop}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.387 \end{prop} Your command was ignored. Type I to replace it with another command, or to continue without it. 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.389 B efore presenting the proof, we will mention that the above You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no B in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! ! Undefined control sequence. l.390 proposition might\edit {What do you mean ``might''? Is it or The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no W in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ? in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no ? in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no O in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no U in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. l.400 ...n K^*$, then any other such solution\edit {What The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no W in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ? in font nullfont! ! Undefined control sequence. l.401 do you mean by ``such solution''?}\edit {You should formalize The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no Y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. l.405 ...l point $P$. Therefore, the question\edit {of} computing The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! ! Undefined control sequence. l.406 Mordell-Weil group\edit {groups. Also, you are only really computing The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no O in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 389--416 [][] [] Overfull \hbox (22.16666pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 E=K$ [] Overfull \hbox (11.69363pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 f[]$ [] Overfull \hbox (22.29402pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]$ [] Overfull \hbox (12.39168pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 Q[]$ [] Overfull \hbox (12.39168pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 Q[]$ [] Overfull \hbox (39.22218pt too wide) in paragraph at lines 389--416 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b[]; b[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 2 [] Overfull \hbox (36.36343pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S; m\OT1/cmr/m/n/10 )$ [] Overfull \hbox (4.29166pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 b$ [] Overfull \hbox (32.01971pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 b[]z[] \OT1/cmr/m/n/10 = [] Overfull \hbox (31.1833pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 P\OT1/cmr/m/n/10 )$ [] Overfull \hbox (32.01971pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 b[]z[] \OT1/cmr/m/n/10 = [] Overfull \hbox (31.1833pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 P\OT1/cmr/m/n/10 )$ [] Overfull \hbox (41.63779pt too wide) in paragraph at lines 389--416 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 P; z[]; z[]\OT1/cmr/m/n/10 )$ [] Overfull \hbox (17.25343pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 P \OMS/cmsy/m/n/10 2 [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (32.16206pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 z[]; z[] \OMS/cmsy/m/n/10 2 [] Overfull \hbox (13.80556pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 K[]$ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 K$ [] Overfull \hbox (7.80904pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 P$ [] Overfull \hbox (63.66904pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )\OML/cmm/ m/it/10 =mE\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (59.80556pt too wide) in paragraph at lines 389--416 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x; 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y\OT1/cmr/m/n/10 )$ [] Overfull \hbox (32.01971pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 b[]z[] \OT1/cmr/m/n/10 = [] Overfull \hbox (38.79555pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x; y\OT1/cmr/m/n/10 )$ [] Overfull \hbox (13.79991pt too wide) in paragraph at lines 389--416 \OT1/cmr/m/it/10 ho- [] Overfull \hbox (16.86656pt too wide) in paragraph at lines 389--416 \OT1/cmr/m/it/10 mo- [] Overfull \hbox (12.77766pt too wide) in paragraph at lines 389--416 \OT1/cmr/m/it/10 ge- [] Overfull \hbox (24.27762pt too wide) in paragraph at lines 389--416 \OT1/cmr/m/it/10 neous [] Overfull \hbox (22.48871pt too wide) in paragraph at lines 389--416 \OT1/cmr/m/it/10 space [] Overfull \hbox (22.16666pt too wide) in paragraph at lines 389--416 \OML/cmm/m/it/10 E=K$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.417 \verb "Proof of Proposition 1.2:" Consider the element $\beta = You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.428 F or the second part of the proposition, recall [Sil-1, III.\S8] You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no F in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no [ in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no 8 in font nullfont! Missing character: There is no ] in font nullfont! 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Missing character: There is no g in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! ! Undefined control sequence. \ds l.430 other words, $\ds e_m(P, Q) := \frac{g_Q(X + P)}{g_Q(X)}$ (the The control sequence at the end of the top line of your error message was never \def'ed. 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Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 428--434 [][] [] Overfull \hbox (4.44447pt too wide) in paragraph at lines 428--434 \OMS/cmsy/m/n/10 x [] Overfull \hbox (11.69363pt too wide) in paragraph at lines 428--434 \OML/cmm/m/it/10 f[]$ [] Overfull \hbox (11.56746pt too wide) in paragraph at lines 428--434 \OML/cmm/m/it/10 g[]$ [] Overfull \hbox (52.41135pt too wide) in paragraph at lines 428--434 \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 P; Q\OT1/cmr/m/n/10 ) := [] Overfull \hbox (37.87122pt too wide) in paragraph at lines 428--434 []$ [] Overfull \hbox (9.06943pt too wide) in paragraph at lines 428--434 \OML/cmm/m/it/10 X$ [] Overfull \hbox (20.05899pt too wide) in paragraph at lines 428--434 \OML/cmm/m/it/10 P[] \OMS/cmsy/m/n/10 2 [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 428--434 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 ([])$ [] Overfull \hbox (35.50581pt too wide) in paragraph at lines 428--434 \OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]\OML/cmm/m/it/10 P[] \OT1/c mr/m/n/10 = [] Overfull \hbox (7.80904pt too wide) in paragraph at lines 428--434 \OML/cmm/m/it/10 P$ [] Overfull \hbox (4.29166pt too wide) in paragraph at lines 428--434 \OML/cmm/m/it/10 b$ [] Overfull \hbox (12.25238pt too wide) in paragraph at lines 428--434 \OML/cmm/m/it/10 e[]$ [] Overfull \hbox (19.62494pt too wide) in paragraph at lines 428--434 \OML/cmm/m/it/10 X \OT1/cmr/m/n/10 = [] Overfull \hbox (10.6146pt too wide) in paragraph at lines 428--434 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (198.0087pt too wide) detected at line 437 [] \OT1/cmr/m/n/10 = \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 P[] \OMS/cmsy/m/n/10 \OML/cmm/m/it/10 P[]; Q\OT1/cmr/m/n/10 ) = [] = []\OML/cmm/m /it/10 : [] Missing character: There is no B in font nullfont! 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Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! 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Missing character: There is no H in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. 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Undefined control sequence. l.446 \section {Applications and Complete 2-descent} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.446 \section{A pplications and Complete 2-descent} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no A in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 446--447 [][] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.448 O ur discussion in section 3 will not be complete without an You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no O in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no 3 in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! 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Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! ! Undefined control sequence. l.455 equations.\edit {Note here that the case $m>2$ can also be The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 448--458 [][] [] Overfull \hbox (19.33565pt too wide) in paragraph at lines 448--458 \OML/cmm/m/it/10 m \OT1/cmr/m/n/10 = [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 448--458 \OT1/cmr/m/n/10 2$ [] Overfull \hbox (19.33565pt too wide) in paragraph at lines 448--458 \OML/cmm/m/it/10 m > [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 448--458 \OT1/cmr/m/n/10 2$ [] Overfull \hbox (19.33565pt too wide) in paragraph at lines 448--458 \OML/cmm/m/it/10 m \OT1/cmr/m/n/10 = [] Overfull \hbox (23.88887pt too wide) in paragraph at lines 448--458 \OT1/cmr/m/n/10 3\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 4\OML/cmm/m/it/10 ; \OT1/cm r/m/n/10 5$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.459 F irst, take a Weierstrass equation for $E$ of the form You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no F in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 459--460 [][] [] Overfull \hbox (7.95831pt too wide) in paragraph at lines 459--460 \OML/cmm/m/it/10 E$ [] Overfull \hbox (117.09854pt too wide) detected at line 462 \OML/cmm/m/it/10 y[] \OT1/cmr/m/n/10 = (\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 \ OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 )(\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 \OML /cmm/m/it/10 e[]\OT1/cmr/m/n/10 )(\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 \OML/cm m/m/it/10 e[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 : [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.464 T he 2-torsion point in $E$ are $0$ and $Q_i = (e_i, 0)$ for $i = You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no [ in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no ] in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 464--470 [][] [] Overfull \hbox (7.95831pt too wide) in paragraph at lines 464--470 \OML/cmm/m/it/10 E$ [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 464--470 \OT1/cmr/m/n/10 0$ [] Overfull \hbox (21.79034pt too wide) in paragraph at lines 464--470 \OML/cmm/m/it/10 Q[] \OT1/cmr/m/n/10 = [] Overfull \hbox (25.20778pt too wide) in paragraph at lines 464--470 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[]; \OT1/cmr/m/n/10 0)$ [] Overfull \hbox (14.00064pt too wide) in paragraph at lines 464--470 \OML/cmm/m/it/10 i \OT1/cmr/m/n/10 = [] Overfull \hbox (23.88887pt too wide) in paragraph at lines 464--470 \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 2\OML/cmm/m/it/10 ; \OT1/cm r/m/n/10 3$ [] Overfull \hbox (14.86148pt too wide) in paragraph at lines 464--470 \OML/cmm/m/it/10 f[]$ [] Overfull \hbox (14.7353pt too wide) in paragraph at lines 464--470 \OML/cmm/m/it/10 g[]$ [] Overfull \hbox (25.41699pt too wide) in paragraph at lines 464--470 \OML/cmm/m/it/10 f[] \OT1/cmr/m/n/10 = [] Overfull \hbox (13.49307pt too wide) in paragraph at lines 464--470 \OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 [] Overfull \hbox (7.98555pt too wide) in paragraph at lines 464--470 \OML/cmm/m/it/10 e[]$ [] Overfull \hbox (33.1948pt too wide) in paragraph at lines 464--470 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 ) = [] Overfull \hbox (31.79045pt too wide) in paragraph at lines 464--470 \OT1/cmr/m/n/10 2(\OML/cmm/m/it/10 Q[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 [] Overfull \hbox (17.77783pt too wide) in paragraph at lines 464--470 \OT1/cmr/m/n/10 2(0)$ [] Overfull \hbox (309.0245pt too wide) detected at line 473 \OML/cmm/m/it/10 x \OMS/cmsy/m/n/10  \OT1/cmr/m/n/10 [2] \OMS/cmsy/m/n/10 \O ML/cmm/m/it/10 e[] \OT1/cmr/m/n/10 = (\OML/cmm/m/it/10 x[] \OMS/cmsy/m/n/10 \ OT1/cmr/m/n/10 2\OML/cmm/m/it/10 e[]x \OMS/cmsy/m/n/10 \OT1/cmr/m/n/10 2\OML/ cmm/m/it/10 e[] \OT1/cmr/m/n/10 + 2(\OML/cmm/m/it/10 e[] \OT1/cmr/m/n/10 + \OML /cmm/m/it/10 e[] \OT1/cmr/m/n/10 + \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 )\OML/cm m/m/it/10 e[] \OMS/cmsy/m/n/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[]e[] \OT1/ cmr/m/n/10 + \OML/cmm/m/it/10 e[]e[] \OT1/cmr/m/n/10 + \OML/cmm/m/it/10 e[]e[]\ OT1/cmr/m/n/10 ))[]\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 (2\OML/cmm/m/it/10 y\OT1/c mr/m/n/10 )[]\OML/cmm/m/it/10 ; [] Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! ! Undefined control sequence. l.474 so\edit {It is bizarre that you did not use frac for the The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no I in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ! in font nullfont! Missing character: There is no ! in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! ! Undefined control sequence. \ds l.476 it!!} we can set $\ds g_{Q_i} = \frac{(x^2 - 2e_ix - 2e_i^2 + 2(e_1 + The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (25.29082pt too wide) in paragraph at lines 473--480 \OML/cmm/m/it/10 g[] \OT1/cmr/m/n/10 = [] Overfull \hbox (186.41876pt too wide) in paragraph at lines 473--480 []$ [] Overfull \hbox (14.86148pt too wide) in paragraph at lines 473--480 \OML/cmm/m/it/10 f[]$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.481 F ix $(b_1, b_2) \in K(S, 2) \times K(S, 2)$. To check whether You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no F in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! ! Undefined control sequence. l.488 the first one, one\edit {change ``one'' to ``we''; also display The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ! in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! ! Undefined control sequence. \ds l.492 ... z_1, z_2$ are non-zero, we consider $\ds z_3 = The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no * in font nullfont! Missing character: There is no * in font nullfont! Missing character: There is no * in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no * in font nullfont! Missing character: There is no * in font nullfont! Missing character: There is no * in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 481--496 [][] [] Overfull \hbox (39.22218pt too wide) in paragraph at lines 481--496 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b[]; b[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 2 [] Overfull \hbox (40.36111pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S; \OT1/cmr/m/n/10 2) \OMS/ cmsy/m/n/10  [] Overfull \hbox (32.58331pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S; \OT1/cmr/m/n/10 2)$ [] Overfull \hbox (29.77779pt too wide) in paragraph at lines 481--496 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b[]; b[]\OT1/cmr/m/n/10 )$ [] Overfull \hbox (4.29166pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 b$ [] Overfull \hbox (20.30324pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 y[] \OT1/cmr/m/n/10 = [] Overfull \hbox (17.38197pt too wide) in paragraph at lines 481--496 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 [] Overfull \hbox (30.41327pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 )(\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 [] Overfull \hbox (30.41327pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 )(\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 [] Overfull \hbox (13.0313pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 )$ [] Overfull \hbox (28.90973pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 b[]z[] \OT1/cmr/m/n/10 = [] Overfull \hbox (13.49307pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 [] Overfull \hbox (9.1424pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 e[]$ [] Overfull \hbox (28.90973pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 b[]z[] \OT1/cmr/m/n/10 = [] Overfull \hbox (13.49307pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 [] Overfull \hbox (9.1424pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 e[]$ [] Overfull \hbox (59.80556pt too wide) in paragraph at lines 481--496 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x; y; z[]; z[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n /10 2 [] Overfull \hbox (16.98613pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 K \OMS/cmsy/m/n/10  [] Overfull \hbox (16.98613pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 K \OMS/cmsy/m/n/10  [] Overfull \hbox (21.58336pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 K[] \OMS/cmsy/m/n/10  [] Overfull \hbox (13.80556pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 K[]$ [] Overfull \hbox (12.39168pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 Q[]$ [] Overfull \hbox (12.39168pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 Q[]$ [] Overfull \hbox (18.5139pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [2]$ [] Overfull \hbox (20.30324pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 y[] \OT1/cmr/m/n/10 = [] Overfull \hbox (17.38197pt too wide) in paragraph at lines 481--496 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 [] Overfull \hbox (49.73973pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 b[]b[]z[]z[]$ [] Overfull \hbox (49.16206pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 b[]; b[]; z[]; z[]$ [] Overfull \hbox (19.69214pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 z[] \OT1/cmr/m/n/10 = [] Overfull \hbox (32.68594pt too wide) in paragraph at lines 481--496 []$ [] Overfull \hbox (37.68752pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 b[]b[]z[] \OT1/cmr/m/n/10 = [] Overfull \hbox (13.49307pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 [] Overfull \hbox (9.1424pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 e[]$ [] Overfull \hbox (28.90973pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 b[]z[] \OT1/cmr/m/n/10 = [] Overfull \hbox (13.49307pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 [] Overfull \hbox (9.1424pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 e[]$ [] Overfull \hbox (28.90973pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 b[]z[] \OT1/cmr/m/n/10 = [] Overfull \hbox (13.49307pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 [] Overfull \hbox (9.1424pt too wide) in paragraph at lines 481--496 \OML/cmm/m/it/10 e[]$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.497 W e can summarize the whole argument in the following You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! ! LaTeX Error: Environment thm undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.498 \begin{thm} [Complete 2-descent] Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no [ in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ] in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 497--501 [][] [] Overfull \hbox (22.16666pt too wide) in paragraph at lines 497--501 \OML/cmm/m/it/10 E=K$ [] Overfull \hbox (148.18115pt too wide) detected at line 503 \OML/cmm/m/it/10 y[] \OT1/cmr/m/n/10 = (\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 \ OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 )(\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 \OML /cmm/m/it/10 e[]\OT1/cmr/m/n/10 )(\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 \OML/cm m/m/it/10 e[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; e[] \OMS/cmsy/m/n/10 2 \OML/ cmm/m/it/10 K [] Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Overfull \hbox (6.70831pt too wide) in paragraph at lines 503--507 \OML/cmm/m/it/10 S$ [] Overfull \hbox (7.95831pt too wide) in paragraph at lines 503--507 \OML/cmm/m/it/10 E$ [] ! Undefined control sequence. l.508 E(K) / 2E(K) \ra K(S, 2) \times K(S, 2), The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Overfull \hbox (135.61113pt too wide) detected at line 509 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )\OML/cmm/ m/it/10 =\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K \OT1/cmr/m/n/10 )\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S; \OT1/cm r/m/n/10 2) \OMS/cmsy/m/n/10  \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 (\OML/cmm/m/it /10 S; \OT1/cmr/m/n/10 2)\OML/cmm/m/it/10 ; [] Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no 3 in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! ! Undefined control sequence. l.513 (x(P) - e_1, x(P) - e_2) & \text {if $x(P) \ne e_1, e_2$},\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing } inserted. } l.513 (x(P) - e_1, x(P) - e_2) & \text{if $ x(P) \ne e_1, e_2$},\\ I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Missing character: There is no x in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no P in font nullfont! Missing character: There is no ) in font nullfont! ! Missing $ inserted. $ l.513 ...) - e_1, x(P) - e_2) & \text{if $x(P) \ne e_1, e_2$},\\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing \cr inserted. \cr l.513 ...P) - e_2) & \text{if $x(P) \ne e_1, e_2$} ,\\ I'm guessing that you meant to end an alignment here. ! Missing $ inserted. $ l.513 ...P) - e_2) & \text{if $x(P) \ne e_1, e_2$} ,\\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.513 ...) - e_2) & \text{if $x(P) \ne e_1, e_2$}, \\ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no , in font nullfont! ! Misplaced \cr. \reserved@c ->\ifnum 0=`{\fi }${}\cr l.514 ( (e_1 - e_3)/(e_1 - e_2), e_1 - e_2) & \text{if $x(P) = e_1$},\\ I can't figure out why you would want to use a tab mark or \cr or \span just now. If something like a right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. Missing character: There is no ( in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no e in font nullfont! ! Missing $ inserted. $ l.514 ((e_ 1 - e_3)/(e_1 - e_2), e_1 - e_2) & \text{if $x(P) = e_1$},\\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Misplaced alignment tab character &. l.514 ((e_1 - e_3)/(e_1 - e_2), e_1 - e_2) & \text{if $x(P) = e_1$},\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.514 ..._1 - e_3)/(e_1 - e_2), e_1 - e_2) & \text {if $x(P) = e_1$},\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing } inserted. } l.514 ...e_3)/(e_1 - e_2), e_1 - e_2) & \text{if $ x(P) = e_1$},\\ I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Missing character: There is no x in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no P in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no = in font nullfont! Missing character: There is no e in font nullfont! ! Missing $ inserted. $ l.514 ... - e_2), e_1 - e_2) & \text{if $x(P) = e_ 1$},\\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. Overfull \hbox (32.26393pt too wide) in paragraph at lines 513--514 [][]$[]$$[] \OMS/cmsy/m/n/10 [] Overfull \hbox (38.84041pt too wide) in paragraph at lines 513--514 \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 (\OML/cm m/m/it/10 e[] \OMS/cmsy/m/n/10 [] Overfull \hbox (34.3959pt too wide) in paragraph at lines 513--514 \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; e[] \OMS/cmsy/m/n/10 [] Overfull \hbox (22.44868pt too wide) in paragraph at lines 513--514 \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 )[]$ [] Overfull \hbox (4.48613pt too wide) in paragraph at lines 513--514 []$ [] ! Missing \endgroup inserted. \endgroup l.514 ...), e_1 - e_2) & \text{if $x(P) = e_1$},\\ I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing \right. inserted. \right . l.514 ...), e_1 - e_2) & \text{if $x(P) = e_1$},\\ I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Display math should end with $$. { l.514 ...), e_1 - e_2) & \text{if $x(P) = e_1$},\\ The `$' that I just saw supposedly matches a previous `$$'. So I shall assume that you typed `$$' both times. Overfull \hbox (178.15538pt too wide) detected at line 514 \OML/cmm/m/it/10 P \OMS/cmsy/m/n/10 7! [] [] ! Misplaced \cr. \reserved@c ->\ifnum 0=`{\fi }${}\cr l.515 ( e_2 - e_1, (e_2 - e_3) / (e_2 - e_1)) & \text{if $x(P) = e_2$},\\ I can't figure out why you would want to use a tab mark or \cr or \span just now. If something like a right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. l.515 (e_2 - e_1, (e_2 - e_3) / (e_2 - e_1)) & \text{if $x(P) = e_2$},\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.515 ...- e_1, (e_2 - e_3) / (e_2 - e_1)) & \text {if $x(P) = e_2$},\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing } inserted. } l.515 ..., (e_2 - e_3) / (e_2 - e_1)) & \text{if $ x(P) = e_2$},\\ I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Missing character: There is no x in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no P in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no = in font nullfont! Missing character: There is no e in font nullfont! ! Missing $ inserted. $ l.515 ...e_3) / (e_2 - e_1)) & \text{if $x(P) = e_ 2$},\\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra }, or forgotten \endgroup. l.515 ...) / (e_2 - e_1)) & \text{if $x(P) = e_2$} ,\\ I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. Missing character: There is no , in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no ) in font nullfont! ! Misplaced alignment tab character &. l.516 (1, 1) & \text{if $P = O$}. I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.516 (1, 1) & \text {if $P = O$}. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! ! Misplaced \crcr. \endarray ->\crcr \egroup \egroup l.517 \end{array} I can't figure out why you would want to use a tab mark or \cr or \span just now. If something like a right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Extra }, or forgotten \endgroup. \endarray ->\crcr \egroup \egroup l.517 \end{array} I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Extra }, or forgotten \endgroup. \endarray ->\crcr \egroup \egroup l.517 \end{array} I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! LaTeX Error: \begin{document} ended by \end{array}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.517 \end{array} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.518 \right . \] I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra \right. l.518 \right. \] I'm ignoring a \right that had no matching \left. ! LaTeX Error: Bad math environment delimiter. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.518 \right. \] Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.519 I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. Overfull \hbox (20.8091pt too wide) in paragraph at lines 514--519 $[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[] \OMS/cmsy/m/n/10 [] Overfull \hbox (34.3959pt too wide) in paragraph at lines 514--519 \OML/cmm/m/it/10 e[]; \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[] \OMS/cmsy/m/n/10 [] Overfull \hbox (38.84041pt too wide) in paragraph at lines 514--519 \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 (\OML/cm m/m/it/10 e[] \OMS/cmsy/m/n/10 [] Overfull \hbox (26.33759pt too wide) in paragraph at lines 514--519 \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 ))[]$ [] Overfull \hbox (4.48613pt too wide) in paragraph at lines 514--519 []$ [] Overfull \hbox (18.36455pt too wide) in paragraph at lines 514--519 \OML/cmm/m/it/10 P \OT1/cmr/m/n/10 = [] Overfull \hbox (7.90555pt too wide) in paragraph at lines 514--519 \OML/cmm/m/it/10 O$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.520 I f $(b_1, b_2) \in K(S, 2) \times K(S, 2)$ is not in the image of You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{thm}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.528 \end{thm} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Extra \endgroup. \endgroup l.528 \end{thm} Things are pretty mixed up, but I think the worst is over. Overfull \hbox (20.0pt too wide) in paragraph at lines 520--529 [][] [] Overfull \hbox (39.22218pt too wide) in paragraph at lines 520--529 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b[]; b[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 2 [] Overfull \hbox (40.36111pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S; \OT1/cmr/m/n/10 2) \OMS/ cmsy/m/n/10  [] Overfull \hbox (32.58331pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 K\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 S; \OT1/cmr/m/n/10 2)$ [] Overfull \hbox (7.90555pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 O$ [] Overfull \hbox (26.36462pt too wide) in paragraph at lines 520--529 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[]; \OT1/cmr/m/n/10 0)$ [] Overfull \hbox (26.36462pt too wide) in paragraph at lines 520--529 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[]; \OT1/cmr/m/n/10 0)$ [] Overfull \hbox (29.77779pt too wide) in paragraph at lines 520--529 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b[]; b[]\OT1/cmr/m/n/10 )$ [] Overfull \hbox (17.25343pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 P \OMS/cmsy/m/n/10 2 [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 K$ [] Overfull \hbox (26.13202pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 b[]z[] \OMS/cmsy/m/n/10 [] Overfull \hbox (28.90973pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 b[]z[] \OT1/cmr/m/n/10 = [] Overfull \hbox (16.9202pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 e[] \OMS/cmsy/m/n/10 [] Overfull \hbox (9.1424pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 e[]$ [] Overfull \hbox (26.13202pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 b[]z[] \OMS/cmsy/m/n/10 [] Overfull \hbox (37.68752pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 b[]b[]z[] \OT1/cmr/m/n/10 = [] Overfull \hbox (16.9202pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 e[] \OMS/cmsy/m/n/10 [] Overfull \hbox (9.1424pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 e[]$ [] Overfull \hbox (53.5209pt too wide) in paragraph at lines 520--529 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 z[]; z[]; z[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/ 10 2 [] Overfull \hbox (21.58336pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 K[] \OMS/cmsy/m/n/10  [] Overfull \hbox (21.58336pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 K[] \OMS/cmsy/m/n/10  [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 K$ [] Overfull \hbox (63.66904pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )\OML/cmm/ m/it/10 =mE\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (31.85762pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 P\OT1/cmr/m/n/10 ) = [] Overfull \hbox (26.13202pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 b[]z[] \OT1/cmr/m/n/10 + [] Overfull \hbox (9.1424pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 e[]$ [] Overfull \hbox (31.40395pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 P\OT1/cmr/m/n/10 ) = [] Overfull \hbox (44.96545pt too wide) in paragraph at lines 520--529 \OML/cmm/m/it/10 b[]b[]z[]z[]z[]$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.530 F inally, let us illustrate the technique of 2-descent with a You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no F in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no : in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 530--532 [][] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.533 $ \ $\\ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! Undefined control sequence. \bf l.534 \textit{\bf Example:} The goal is to compute the weak Mordell-Weil The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! ! Undefined control sequence. l.535 group $E(\Q ) / 2E(\Q)$ for the curve\edit{Instead of doing The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.535 group $E(\Q) / 2E(\Q )$ for the curve\edit{Instead of doing The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! ! Undefined control sequence. l.535 group $E(\Q) / 2E(\Q)$ for the curve\edit {Instead of doing The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ? in font nullfont! Missing character: There is no O in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no ? in font nullfont! Missing character: There is no O in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 533--540 [][]$ $ [] Overfull \hbox (40.04422pt too wide) in paragraph at lines 533--540 \OT1/cmr/m/it/10 Example: [] Overfull \hbox (41.47226pt too wide) in paragraph at lines 533--540 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 ()\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 2\OML/cmm /m/it/10 E\OT1/cmr/m/n/10 ()$ [] Overfull \hbox (16.55786pt too wide) in paragraph at lines 533--540 \OML/cmm/m/it/10 n \OT1/cmr/m/n/10 = [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 533--540 \OT1/cmr/m/n/10 3$ [] Overfull \hbox (16.55786pt too wide) in paragraph at lines 533--540 \OML/cmm/m/it/10 n \OT1/cmr/m/n/10 = [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 533--540 \OT1/cmr/m/n/10 1$ [] Overfull \hbox (165.41423pt too wide) detected at line 542 \OML/cmm/m/it/10 E \OT1/cmr/m/n/10 : \OML/cmm/m/it/10 y[] \OT1/cmr/m/n/10 = \OM L/cmm/m/it/10 x[] \OMS/cmsy/m/n/10 \OT1/cmr/m/n/10 7\OML/cmm/m/it/10 x[] \OT1 /cmr/m/n/10 + 5\OML/cmm/m/it/10 x \OT1/cmr/m/n/10 = \OML/cmm/m/it/10 x\OT1/cmr/ m/n/10 (\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 \OT1/cmr/m/n/10 2)(\OML/cmm/m/it/ 10 x \OMS/cmsy/m/n/10 \OT1/cmr/m/n/10 5)\OML/cmm/m/it/10 : [] Missing character: There is no F in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (18.88887pt too wide) in paragraph at lines 542--546 \OT1/cmr/m/n/10  = [] Overfull \hbox (12.26393pt too wide) in paragraph at lines 542--546 \OT1/cmr/m/n/10 2[] \OMS/cmsy/m/n/10  [] Overfull \hbox (12.26393pt too wide) in paragraph at lines 542--546 \OT1/cmr/m/n/10 3[] \OMS/cmsy/m/n/10  [] Overfull \hbox (9.48615pt too wide) in paragraph at lines 542--546 \OT1/cmr/m/n/10 5[]$ [] ! Undefined control sequence. l.547 \section {Definition of Selmer and Shafarevich-Tate Groups} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.547 \section{D efinition of Selmer and Shafarevich-Tate Groups} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no D in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 547--551 [][] [] Overfull \hbox (63.66904pt too wide) in paragraph at lines 547--551 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )\OML/cmm/ m/it/10 =mE\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.552 I n the previous section, we obtained the Kummer sequence out of You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no K in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no : in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! ! Undefined control sequence. l.554 ...neral setting: suppose that $\phi : E \ra E'$ is a The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! ! Undefined control sequence. l.556 exact sequence\edit {In the first term, $E(K)$ should The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 552--558 [][] [] Overfull \hbox (11.51384pt too wide) in paragraph at lines 552--558 \OML/cmm/m/it/10  \OT1/cmr/m/n/10 : [] Overfull \hbox (18.72218pt too wide) in paragraph at lines 552--558 \OML/cmm/m/it/10 EE[]$ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 552--558 \OML/cmm/m/it/10 K$ [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 552--558 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (38.68059pt too wide) in paragraph at lines 552--558 \OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/ m/it/10 K\OT1/cmr/m/n/10 ))$ [] ! LaTeX Error: Environment CD undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.559 \begin{CD} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Undefined control sequence. l.560 0 \ra E[\phi] @>>> E @>\phi>> E' \ra 0. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.560 0 \ra E[\phi] @>>> E @>\phi>> E' \ra 0. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: \begin{document} ended by \end{CD}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.561 \end{CD} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (135.81924pt too wide) detected at line 562 \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 \OT1/cmr/ m/n/10 ]@ \OML/cmm/m/it/10 >>> E\OT1/cmr/m/n/10 @ \OML/cmm/m/it/10 >  >> E[]\O T1/cmr/m/n/10 0\OML/cmm/m/it/10 : [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.564 I n precisely the same way as for the case $E' = E$ and $\phi = You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 564--566 [][] [] Overfull \hbox (21.31938pt too wide) in paragraph at lines 564--566 \OML/cmm/m/it/10 E[] \OT1/cmr/m/n/10 = [] Overfull \hbox (7.95831pt too wide) in paragraph at lines 564--566 \OML/cmm/m/it/10 E$ [] Overfull \hbox (16.51385pt too wide) in paragraph at lines 564--566 \OML/cmm/m/it/10  \OT1/cmr/m/n/10 = [] Overfull \hbox (14.33571pt too wide) in paragraph at lines 564--566 \OT1/cmr/m/n/10 [\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ]$ [] ! LaTeX Error: Environment CD undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.567 \begin{CD} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Undefined control sequence. l.568 0 \ra E'(K) / E(K) @>\delta>> H^1(\Gk, E[\phi]) @>>> H^1(\Gk, The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.568 0 \ra E'(K) / E(K) @>\delta>> H^1(\Gk , E[\phi]) @>>> H^1(\Gk, The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.568 ...@>\delta>> H^1(\Gk, E[\phi]) @>>> H^1(\Gk , The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.569 E(\overline{K}))[\phi] \ra 0 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: \begin{document} ended by \end{CD}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.570 \end{CD} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (259.00328pt too wide) detected at line 571 \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 E[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cm r/m/n/10 )\OML/cmm/m/it/10 =E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )@ \OML/cmm/m/it/10 >  >> H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ; E\OT1/cmr/m/ n/10 [\OML/cmm/m/it/10 \OT1/cmr/m/n/10 ])@ \OML/cmm/m/it/10 >>> H[]\OT1/cmr/m/ n/10 (\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 ([]))[\OML/cmm/m/it/10 \OT1/cmr/m/n/ 10 ]0 [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.573 N ext, we consider a place $\nu$ for the number field $K$. Extend You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no N in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no K in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 573--582 [][] [] Overfull \hbox (5.57639pt too wide) in paragraph at lines 573--582 \OML/cmm/m/it/10 $ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 573--582 \OML/cmm/m/it/10 K$ [] Overfull \hbox (5.57639pt too wide) in paragraph at lines 573--582 \OML/cmm/m/it/10 $ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 573--582 []$ [] Overfull \hbox (19.76384pt too wide) in paragraph at lines 573--582 [] \OMS/cmsy/m/n/10  [] Overfull \hbox (14.18405pt too wide) in paragraph at lines 573--582 [][]$ [] Overfull \hbox (23.8104pt too wide) in paragraph at lines 573--582 \OML/cmm/m/it/10 D[] \OMS/cmsy/m/n/10  [] Overfull \hbox (46.8195pt too wide) in paragraph at lines 573--582 []\OT1/cmr/m/n/10 ([]\OML/cmm/m/it/10 =K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (14.18405pt too wide) in paragraph at lines 573--582 [][]$ [] Overfull \hbox (13.25488pt too wide) in paragraph at lines 573--582 \OML/cmm/m/it/10 D[]$ [] Overfull \hbox (29.92017pt too wide) in paragraph at lines 573--582 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 ([][])$ [] Overfull \hbox (32.72572pt too wide) in paragraph at lines 573--582 \OML/cmm/m/it/10 E[]\OT1/cmr/m/n/10 ([][])$ [] ! Undefined control sequence. l.583 0 \ra E'(K_\nu) / \phi(E(K_\nu)) \ra H^1(D_\nu, E[\phi]) \ra The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.583 0 \ra E'(K_\nu) / \phi(E(K_\nu)) \ra H^1(D_\nu, E[\phi]) \ra The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.583 ...phi(E(K_\nu)) \ra H^1(D_\nu, E[\phi]) \ra The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.584 H^1(D_\nu, E(\overline{K}))[\phi] \ra 0. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Overfull \hbox (226.8363pt too wide) detected at line 585 \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 E[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K[]\OT1/ cmr/m/n/10 )\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 E\OT1/cmr/m/n/ 10 (\OML/cmm/m/it/10 K[]\OT1/cmr/m/n/10 ))\OML/cmm/m/it/10 H[]\OT1/cmr/m/n/10 ( \OML/cmm/m/it/10 D[]; E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 \OT1/cmr/m/n/10 ])\OM L/cmm/m/it/10 H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 D[]; E\OT1/cmr/m/n/10 ([]))[ \OML/cmm/m/it/10 \OT1/cmr/m/n/10 ]0\OML/cmm/m/it/10 : [] Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! ! Undefined control sequence. l.587 ...set E(\overline{K}_\nu)$. But recall\edit {why The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no ? in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no O in font nullfont! Missing character: There is no K in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Overfull \hbox (23.8104pt too wide) in paragraph at lines 585--598 \OML/cmm/m/it/10 D[] \OMS/cmsy/m/n/10  [] Overfull \hbox (46.8195pt too wide) in paragraph at lines 585--598 []\OT1/cmr/m/n/10 ([]\OML/cmm/m/it/10 =K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (35.49995pt too wide) in paragraph at lines 585--598 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 ([]) \OMS/cmsy/m/n/10  [] Overfull \hbox (29.92017pt too wide) in paragraph at lines 585--598 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 ([][])$ [] Overfull \hbox (29.92017pt too wide) in paragraph at lines 585--598 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 ([][])$ [] Overfull \hbox (64.03264pt too wide) in paragraph at lines 585--598 \OML/cmm/m/it/10 H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 D[]; E\OT1/cmr/m/n/10 ([] ))$ [] Overfull \hbox (69.00836pt too wide) in paragraph at lines 585--598 \OML/cmm/m/it/10 H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 D[]; E\OT1/cmr/m/n/10 ([] []))$ [] Overfull \hbox (5.57639pt too wide) in paragraph at lines 585--598 \OML/cmm/m/it/10 $ [] Overfull \hbox (5.57639pt too wide) in paragraph at lines 585--598 \OML/cmm/m/it/10 $ [] ! LaTeX Error: Environment CD undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.599 \begin{CD} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Undefined control sequence. l.600 0 \ra & E'(K) / \phi(E(K)) @>\delta>> The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Misplaced alignment tab character &. l.600 0 \ra & E'(K) / \phi(E(K)) @>\delta>> I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. l.602 ...trm{Gal}(\overline{K} / K), E(K))[\phi] & \ra 0 \\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.602 ...Gal}(\overline{K} / K), E(K))[\phi] & \ra 0 \\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Misplaced alignment tab character &. & l.603 & @VVV @VVV @VVV \\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.604 0 \ra & \prod_{\nu} E'(K_\nu) / \phi(E(K_\nu)) @>\delta>> The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Misplaced alignment tab character &. l.604 0 \ra & \prod_{\nu} E'(K_\nu) / \phi(E(K_\nu)) @>\delta>> I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. l.606 E(K_\nu))[\phi] & \ra 0 I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.606 E(K_\nu))[\phi] & \ra 0 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: \begin{document} ended by \end{CD}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.607 \end{CD} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (819.24231pt too wide) detected at line 608 \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 E[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cm r/m/n/10 )\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 ))@ \OML/cmm/m/it/10 >  >> H[]\OT1/cmr/m/n /10 ([]([]\OML/cmm/m/it/10 =K\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/ 10 [\OML/cmm/m/it/10 \OT1/cmr/m/n/10 ])@ \OML/cmm/m/it/10 >>> H[]\OT1/cmr/m/n/ 10 ([]([]\OML/cmm/m/it/10 =K\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/1 0 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 ))[\OML/cmm/m/it/10 \OT1/cmr/m/n/10 ]0[] @\OML/cmm/m/it/10 VVV\OT1/cmr/m/n/10 @\OML/cmm/m/it/10 VVV\OT1/cmr/m/n/10 @\OML /cmm/m/it/10 VVV[] \OT1/cmr/m/n/10 0 [] \OML/cmm/m/it/10 E[]\OT1/cmr/m/n/10 (\O ML/cmm/m/it/10 K[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 (\OML/cmm /m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K[]\OT1/cmr/m/n/10 ))@ \OML/cmm/m/i t/10 >  >> [] H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 D[]; E\OT1/cmr/m/n/10 [\OML /cmm/m/it/10 \OT1/cmr/m/n/10 ])@ \OML/cmm/m/it/10 >>> [] H[]\OT1/cmr/m/n/10 (\ OML/cmm/m/it/10 D[]; E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K[]\OT1/cmr/m/n/10 ))[\ OML/cmm/m/it/10 \OT1/cmr/m/n/10 ]0 [] Missing character: There is no B in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! ! Undefined control sequence. l.613 and $WC(E / K_\nu)$ respectively.\edit {What do you mean ``change... by''? The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no W in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ? in font nullfont! Missing character: There is no D in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ? in font nullfont! Overfull \hbox (48.64029pt too wide) in paragraph at lines 608--615 \OML/cmm/m/it/10 WC\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 E=K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (80.61113pt too wide) in paragraph at lines 608--615 \OML/cmm/m/it/10 H[]\OT1/cmr/m/n/10 ([]([]\OML/cmm/m/it/10 =K\OT1/cmr/m/n/10 )\ OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 )$ [] Overfull \hbox (48.64029pt too wide) in paragraph at lines 608--615 \OML/cmm/m/it/10 WC\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 E=K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (52.90074pt too wide) in paragraph at lines 608--615 \OML/cmm/m/it/10 WC\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 E=K[]\OT1/cmr/m/n/10 )$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.616 O ur ultimate goal is computing the image of $E'(K) / \phi(E(K))$ in You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no O in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. l.617 $H^1(\Gk , E[\phi])$, which is the same as computing the kernel of the The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! ! Undefined control sequence. l.618 map\edit {The $E$ should be $E(K)[\phi]$. I.e., the $\phi$ goes on the The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ! in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! ! Undefined control sequence. l.619 inside! This is very important.} $H^1(\Gk , E)[\phi]) \ra WC(E / The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.619 ...very important.} $H^1(\Gk, E)[\phi]) \ra WC(E / The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 616--623 [][] [] Overfull \hbox (71.4306pt too wide) in paragraph at lines 616--623 \OML/cmm/m/it/10 E[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )\OML/cm m/m/it/10 =\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/1 0 K\OT1/cmr/m/n/10 ))$ [] Overfull \hbox (45.30554pt too wide) in paragraph at lines 616--623 \OML/cmm/m/it/10 H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 [\OML/ cmm/m/it/10 \OT1/cmr/m/n/10 ])$ [] Overfull \hbox (7.95831pt too wide) in paragraph at lines 616--623 \OML/cmm/m/it/10 E$ [] Overfull \hbox (36.45836pt too wide) in paragraph at lines 616--623 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )[\OML/cmm /m/it/10 \OT1/cmr/m/n/10 ]$ [] Overfull \hbox (5.95834pt too wide) in paragraph at lines 616--623 \OML/cmm/m/it/10 $ [] Overfull \hbox (109.93198pt too wide) in paragraph at lines 616--623 \OML/cmm/m/it/10 H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ; E\OT1/cmr/m/n/10 )[\OML /cmm/m/it/10 \OT1/cmr/m/n/10 ])\OML/cmm/m/it/10 WC\OT1/cmr/m/n/10 (\OML/cmm/m/ it/10 E=K\OT1/cmr/m/n/10 )[\OML/cmm/m/it/10 '\OT1/cmr/m/n/10 ]$ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 616--623 \OML/cmm/m/it/10 K$ [] Overfull \hbox (48.64029pt too wide) in paragraph at lines 616--623 \OML/cmm/m/it/10 WC\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 E=K\OT1/cmr/m/n/10 )$ [] ! LaTeX Error: Environment prop undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.624 \begin{prop} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.625 S uppose that $C / K$ is a homogeneous space for $E / K$. Then $C / K$ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no S in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! ! Undefined control sequence. l.626 represents a\edit {``the'', not ``a''} trivial element of $WC(E / K)$ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{prop}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.628 \end{prop} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 625--629 [][] [] Overfull \hbox (21.79305pt too wide) in paragraph at lines 625--629 \OML/cmm/m/it/10 C=K$ [] Overfull \hbox (22.16666pt too wide) in paragraph at lines 625--629 \OML/cmm/m/it/10 E=K$ [] Overfull \hbox (21.79305pt too wide) in paragraph at lines 625--629 \OML/cmm/m/it/10 C=K$ [] Overfull \hbox (48.64029pt too wide) in paragraph at lines 625--629 \OML/cmm/m/it/10 WC\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 E=K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (7.86249pt too wide) in paragraph at lines 625--629 \OML/cmm/m/it/10 C$ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 625--629 \OML/cmm/m/it/10 K$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.630 \verb "Proof:" One of the directions is easy. Suppose that $C/K$ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no O in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. l.632 $K$-isomorphism $\varphi : E \ra C$. Then $\varphi(0) \in C(K)$, The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (51.49973pt too wide) in paragraph at lines 630--634 [][][]\OT1/cmtt/m/n/10 Proof: [] Overfull \hbox (21.79305pt too wide) in paragraph at lines 630--634 \OML/cmm/m/it/10 C=K$ [] Overfull \hbox (48.64029pt too wide) in paragraph at lines 630--634 \OML/cmm/m/it/10 WC\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 E=K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 630--634 \OML/cmm/m/it/10 K$ [] Overfull \hbox (12.09717pt too wide) in paragraph at lines 630--634 \OML/cmm/m/it/10 ' \OT1/cmr/m/n/10 : [] Overfull \hbox (15.8208pt too wide) in paragraph at lines 630--634 \OML/cmm/m/it/10 EC$ [] Overfull \hbox (28.76389pt too wide) in paragraph at lines 630--634 \OML/cmm/m/it/10 '\OT1/cmr/m/n/10 (0) \OMS/cmsy/m/n/10 2 [] Overfull \hbox (24.84862pt too wide) in paragraph at lines 630--634 \OML/cmm/m/it/10 C\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (24.84862pt too wide) in paragraph at lines 630--634 \OML/cmm/m/it/10 C\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.635 C onversely, suppose that $C(K)$ is non-empty, i.e. $P_0 \in C(K)$. You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no D in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. l.636 Define a morphism $\theta : E \ra C$ by $\theta(Q) = P_0 + Q$. We The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 635--639 [][] [] Overfull \hbox (24.84862pt too wide) in paragraph at lines 635--639 \OML/cmm/m/it/10 C\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (20.35066pt too wide) in paragraph at lines 635--639 \OML/cmm/m/it/10 P[] \OMS/cmsy/m/n/10 2 [] Overfull \hbox (24.84862pt too wide) in paragraph at lines 635--639 \OML/cmm/m/it/10 C\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (10.52773pt too wide) in paragraph at lines 635--639 \OML/cmm/m/it/10  \OT1/cmr/m/n/10 : [] Overfull \hbox (15.8208pt too wide) in paragraph at lines 635--639 \OML/cmm/m/it/10 EC$ [] Overfull \hbox (31.21109pt too wide) in paragraph at lines 635--639 \OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 Q\OT1/cmr/m/n/10 ) = [] Overfull \hbox (18.68407pt too wide) in paragraph at lines 635--639 \OML/cmm/m/it/10 P[] \OT1/cmr/m/n/10 + [] Overfull \hbox (7.90555pt too wide) in paragraph at lines 635--639 \OML/cmm/m/it/10 Q$ [] Overfull \hbox (4.97223pt too wide) in paragraph at lines 635--639 \OML/cmm/m/it/10 $ [] Overfull \hbox (9.20833pt too wide) in paragraph at lines 635--639 \OML/cmm/m/it/10 K$ [] Overfull \hbox (15.5173pt too wide) in paragraph at lines 635--639 \OML/cmm/m/it/10  \OMS/cmsy/m/n/10 2 [] Overfull \hbox (46.8195pt too wide) in paragraph at lines 635--639 []\OT1/cmr/m/n/10 ([]\OML/cmm/m/it/10 =K\OT1/cmr/m/n/10 )$ [] Overfull \hbox (214.18185pt too wide) detected at line 642 \OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 Q\OT1/cmr/m/n/10 )[] = (\OM L/cmm/m/it/10 P[] \OT1/cmr/m/n/10 + \OML/cmm/m/it/10 Q\OT1/cmr/m/n/10 )[] = \OM L/cmm/m/it/10 P[] \OT1/cmr/m/n/10 + \OML/cmm/m/it/10 Q[] \OT1/cmr/m/n/10 = \OML /cmm/m/it/10 P[] \OT1/cmr/m/n/10 + \OML/cmm/m/it/10 Q[] \OT1/cmr/m/n/10 = \OML/ cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 Q[]\OT1/cmr/m/n/10 )\OML/cmm/m/i t/10 : [] Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! ! Undefined control sequence. l.648 \overline{K}(C) \ra \overline{K}(E)$ is an isomorphism of fields. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! ! Undefined control sequence. l.651 $\theta^{-1} : \overline{K}(E) \ra \overline{K}(C)$. This The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. l.652 ...rise to a rational function $\psi : C \ra E$. We The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no [ in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no ] in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no * in font nullfont! Missing character: There is no * in font nullfont! Missing character: There is no * in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no * in font nullfont! Missing character: There is no * in font nullfont! Missing character: There is no * in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: Command \Box not provided in base LaTeX2e. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.655 (***LATER***). $\hfill \Box $ Load the latexsym or the amsfonts package to define this symbol Overfull \hbox (9.20833pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 K$ [] Overfull \hbox (4.97223pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 $ [] Overfull \hbox (7.95831pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 E$ [] Overfull \hbox (7.86249pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 C$ [] Overfull \hbox (17.25343pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 P \OMS/cmsy/m/n/10 2 [] Overfull \hbox (7.86249pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 C$ [] Overfull \hbox (17.34995pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 Q \OMS/cmsy/m/n/10 2 [] Overfull \hbox (7.95831pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 E$ [] Overfull \hbox (31.21109pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 Q\OT1/cmr/m/n/10 ) = [] Overfull \hbox (7.80904pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 P$ [] Overfull \hbox (4.97223pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 $ [] Overfull \hbox (15.12495pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 [] \OT1/cmr/m/n/10 : [] Overfull \hbox (49.79306pt too wide) in paragraph at lines 642--656 []\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 C\OT1/cmr/m/n/10 )[](\OML/cmm/m/it/10 E\OT1 /cmr/m/n/10 )$ [] Overfull \hbox (44.97359pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 [][]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 C\OT1/cmr/m/n/10 ) = [] Overfull \hbox (24.94444pt too wide) in paragraph at lines 642--656 []\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 )$ [] Overfull \hbox (4.97223pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 $ [] Overfull \hbox (21.26387pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 [] \OT1/cmr/m/n/10 : [] Overfull \hbox (49.79306pt too wide) in paragraph at lines 642--656 []\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 )[](\OML/cmm/m/it/10 C\OT1 /cmr/m/n/10 )$ [] Overfull \hbox (12.42819pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 \OT1/cmr/m/n/10 : [] Overfull \hbox (15.8208pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 CE$ [] Overfull \hbox (6.8727pt too wide) in paragraph at lines 642--656 \OML/cmm/m/it/10 $ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.657 A lthough we obtained a simple criteria to check if a principal You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no A in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! 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Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! 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Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no H in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i i