Topics in Algebraic Number Theory

Calendar

Lec #	TOPICS	KEY DATES
1	Course Overview
2	Localization, Examples; Integral Dependence, Integral Closure; Discrete Valuation Rings (Definition)
3	Discrete Valuation Rings (Properties), Dedekind Domains, Unique Factorization of Ideals
4	Fractional Ideals of a Dedekind Domain, Class Group, Finite Extensions of Fields, Norm, Trace, Discriminant	Problem set 1 due
5	Trace and Norm, Separability, Nondegeneracy of the Trace Pairing for a Separable Extension, Extension of Dedekind Domains in the Separable Case
6	Extension of Prime Ideals, Relative Degree, Ramification Degree, The Fundamental Equality, Discriminant	Problem set 2 due
7	Discriminants and Ramification, Norms of Ideals
8	Norm of a Prime Ideal; Properties of Cyclotomic Fields (Prime Power Case)	Problem set 3 due
9	Linearly Disjoint Extensions; Cyclotomic Fields (General Case)
10	Why Quadratic Reciprocity is Now Easy; Real and Complex Embeddings, Lattices	Problem set 4 due
11	Lattices and Ideal Classes, Minkowski’s Theorem, Finiteness of the Class Group; Dirichlet’s Units Theorem
12	Proof of Dirichlet’s Units Theorem	Problem set 5 due
13	Absolute Values; Completions of Fields with Respect to an Absolute Value, Examples; Dichotomy between Archimedean Nonarchimedean Absolute Values; Absolute Values Coming from Discrete Valuation Rings; Normalized Absolute Values (Places), Statement of the Product Formula for Number Fields; Classification of Completions of the Rational Numbers (Ostrowski’s Theorem)
14	In-class Midterm Exam	Problem set 6 due
15	Ostrowski’s Theorem (cont.); Exponential and Logarithm Series; Hensel’s Lemma for Nonarchimedean Absolute Values; Extensions of Nonarchimedean Absolute Values
16	Extension of Nonarchimedean Absolute Values	Detachable midterm exam due Problem set 7 due
17	Classification of Absolute Values on a Number Field; Product Formula for Number Fields; Unramified Extensions	Problem set 8 due
18	Decomposition and Inertia Groups, Frobenius Elements, Artin Symbols
19	Artin Maps for Abelian Extensions; Ray Class Groups; The Artin Reciprocity Law; Proof in the Cyclotomic Case	Problem set 9 due
20	More on Ray Class Groups; Idelic Interpretation
21	Dirichlet Series, Dedekind Zeta Functions, L-series, Dirichlet’s Theorem and Generalizations	Problem set 10 due
22	Chebotarev Density Theorem; Arakelov Class Group
23	Arakelov Class Group (cont.); Local Class Field Theory
24	Local Class Field Theory (cont.); The Adelic Reciprocity Map; The Principal Ideal Theorem
25	Class Field Towers; Complex Multiplication	Take-home final exam due