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\title{Harvard Math 129: Algebraic Number Theory\\
\vspace{2ex}\\
{\Huge\sc Midterm}}
\author{William Stein}
\date{\bf Due: 10:10am in class on Thursday, March 17, 2005}
\begin{document}
\maketitle
\noindent{\em This midterm is worth 25\% of your grade. There are
four problems and this exam is out of 100 points. The
point value of each problem is as indicated.\vspace{2ex}}
\noindent{\bf Rules:} {\em You may use your course notes and textbook, but no other books,
the internet, or people.
You may use math software,
including {\sc Magma} and {\rm PARI}, though I don't think
you will need to for any of these problems, since they are
all conceptual.}
\begin{enumerate}
\item (20 points) Let $R$ be a noetherian integral domain, and let $K=\Frac(R)$
be the field of fractions of $R$. Let $\Kbar$ be an algebraic
closure of $K$. Let $\overline{R}$ be the set of $\alpha \in \Kbar$
such that there is a nonzero monic polynomial $f(x) \in R[x]$ with
$f(\alpha)=0$. Is $\overline{R}$ necessarily a ring? Prove or
give a counterexample.
\item (10 points) Let $\O_K$ be the ring of integers of a number field~$K$,
and suppose $K$ has exactly $2s$ complex embeddings. Prove
that the sign of the discriminant of $\O_K$ is $(-1)^s$.
\item Suppose $K$ is a number field. For any
finite extension~$L$ of $K$, define set-theoretic maps
\begin{align*}
\Psi_L:&\,\, C_K \to C_L, \qquad [I] \mapsto [I \O_L]\\
\Phi_L:&\,\, C_L \to C_K, \qquad [I] \mapsto [I \cap \O_K],
\end{align*}
where $[I]$ denotes the class of the nonzero integral ideal~$I$.
\begin{enumerate}
\item (10 points) Is $\Psi_L$ a group homomorphism? Prove
or give a counterexample.
\item (10 points) Is $\Phi_L$ a group homomorphism? Prove
or give a counterexample.
\item (20 points) Prove that there is a number field $L$ such that
$\Psi_L$ is the~$0$ map, i.e., $\Psi_L$ sends every element of~$C_K$
to the identity of~$C_L$.
[Hint: Use finiteness of $C_K$ in two ways.]
\end{enumerate}
{\small (Note: I wonder, is there always an $L$ such that $\Phi_L$ is the
$0$ map? This question just occured to me while writing
this exam. If you find an answer and tell me the answer,
I'll be very thankful, though this is not part of the
official exam. This question is related to ``visibility of
Mordell-Weil groups'', which I just wrote a paper about.)}
\item
A number field is {\em totally real} if every embedding is real,
i.e., $s=0$, and a number field is {\em totally complex} if every
embedding is complex, i.e., $r=0$.
\begin{enumerate}
\item (15 points) Find with proof the possible degrees of totally real fields.
%(Your answer should be to state a subset of the positive integers and a proof
%that your answer is correct.)
\item (15 points) Find with proof the possible degrees of totally complex fields.
\end{enumerate}
\end{enumerate}
\end{document}
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