**William Stein
(based heavily on works of Swinnerton-Dyer and Cassels)**

*Date:* May 2004

- Contents

- Classical Viewpoint
- Finitely generated abelian groups
- Commutative Algebra
- Rings of Algebraic Integers
- Unique Factorization of Ideals
- Computing
- Factoring Primes
- Chinese Remainder Theorem
- Discrimannts, Norms, and Finiteness of the Class Group
- Computing Class Groups
- Dirichlet's Unit Theorem
- Decomposition and Inertia Groups
- Decomposition Groups and Galois Representations

- Adelic Viewpoint
- Valuations
- Topology and Completeness
- Adic Numbers: The Finite Residue Field Case
- Normed Spaces and Tensor Products
- Extensions and Normalizations of Valuations
- Global Fields and Adeles
- Ideles and Ideals
- Exercises

- Bibliography
- Index
- About this document ...

William Stein 2004-05-06