Introduction to Algebraic Number Theory
By William Stein
What you will learn about:
- Rings of integers of number fields
- Unique factorization of ideals in Dedekind domains
- Structure of the group of units of the ring of integers
- Finiteness of the class group of the ring of integers
- Decomposition and inertia groups, Frobenius elements
- Local fields
- The product formula
- Weak approximation
- Discriminant and different
- Idèles and adèles
- Quadratic fields, biquadratic fields
- Cyclotomic fields and Fermat's last theorem
- How to use a computer to compute with many of the above objects (both algorithms and actual use of PARI and MAGMA)
We will probably not have time to discuss Dirichlet L-functions or class field theory.
- Homework: 50%
- Final project: 30%
- One-day take-home final: 20% (as always, the take-home nature of the final is subject to university approval)
Prerequisites: At the beginning of the course I will quickly review the following topics, which I assume you have already seen:
- Group theory: subgroups, topology, quotients, actions.
- Commutatives rings: ideals, quotients, product of ideals, tensor products
- Galois theory: finite extensions of Q, Galois groups, compositums, Galois closure of a field, subgroups of Galois group correspond to intermediate fields, algebraic closure of Q, complex numbers are algebraically closed, norms and traces.
- Finitely generated abelian groups: structure theorems, subgroup of finitely generated is finitely generated.
- Elementary number theory: It will also be helpful (but not essential) if you've seen some elementary number theory, up to binary quadratic forms and quadratic reciprocity.
If you haven't seen some of these topics, do not be scared off. It just means you should do some extra background reading and exercises in order to get up to speed. See me for advice about what to read. The material about finitely generated abelian groups is in appendix A.1 to Swinnerton-Dyer's book. Artin's Algebra is a good reference for everything, except elementary number theory, which you can read about in my book.