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  Inclass 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, Lseries, 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  Takehome final exam due 
Calendar
Instructor:  
Course Number: 

Departments:  
As Taught In:  Spring 2006 
Level: 
Graduate

Learning Resource Types
assignment
Problem Sets
grading
Exams