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
The product formula
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.
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.