18.786 | Spring 2006 | Graduate

Topics in Algebraic Number Theory


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

Course Info

As Taught In
Spring 2006
Learning Resource Types
Problem Sets