- Week
1: The ring of integers in a number field is a
Dedekind domain. Unique factorization of ideals.
- Week 2: Unique factorization of ideals
(continued), embedding into the complex numbers,
Minkowski's theorem on the geometry of numbers.
- Week
3: Finiteness of class numbers, Dirichlet Unit
Theorem. Extension of fields and decomposition of
prime ideals.
- Week
4: Decomposition of a prime ideal and the
decomposition of the minimal polynomial. Quadratic
fields, class number of Q(\sqrt{-6}) and
Q(\sqrt{-26}), ring of integers of Q(cubic root of
2).
- Week
5: Cyclotomic fields, the first case of Fermat's
Last Theorem for regular primes.
- Week
6: Valuations, completions, local fields. (weak
approximation, convergence in Q_p, Z_p = lim
Z/p^nZ, Q_p is locally compact, Hensel's lemma)
- Week
7: L \otimes K_p = L_{P_1} \oplus \cdots \oplus
L_{P_g}.
- Week
8: A prime ramifies if and only if it divides the
different.
- Week
9: A/K is compact. A^1/K^\times is compact.
- Week
10: Duality, Fourier transform, Poisson summation
formula.
- Week
11: Analytic continuation and functional equation of
global zeta integrals Z(s, \chi, f)
- Week
12: Break.
- Week
13: Analytic continuation and functional equation of
local zeta integrals. Dedekind zeta function,
residue at the pole.
- Week
14: Class Number Formula for imaginary quadratic
fields.
- Week
15: Density. Dirichlet's theorem, density of totally
decomposed ideals;
Frobenius
conjugacy class, Artin L-functions.
- Week
16: Chebotarev Density Theorem.