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Contents
Contents
Index
A B
RIEF
I
NTRODUCTION TO
C
LASSICAL AND
A
DELIC
A
LGEBRAIC
N
UMBER
T
HEORY
William Stein
(based heavily on works of Swinnerton-Dyer and Cassels)
Date:
May 2004
Contents
Preface
Introduction
Mathematical background I assume you have
What is algebraic number theory?
Topics in this book
Some applications of algebraic number theory
Classical Viewpoint
Finitely generated abelian groups
Commutative Algebra
Noetherian Rings and Modules
is Noetherian
Rings of Algebraic Integers
Rings of Algebraic Integers
Norms and Traces
Unique Factorization of Ideals
Dedekind Domains
Computing
Algorithms for Algebraic Number Theory
Using
Smith Normal Form
and Number Fields
Rings of integers
Ideals
Factoring Primes
Factoring Primes
A Method for Factoring that Often Works
A Method for Factoring that Always Works
Essential Discriminant Divisors
Chinese Remainder Theorem
The Chinese Remainder Theorem
Discrimannts, Norms, and Finiteness of the Class Group
Preliminary Remarks
Discriminants
Norms of Ideals
Finiteness of the Class Group via Geometry of Numbers
An Open Problem
Computing Class Groups
Remarks on Computing the Class Group
Dirichlet's Unit Theorem
The Group of Units
Finishing the proof of Dirichlet's Unit Theorem
Some Examples of Units in Number Fields
Preview
Decomposition and Inertia Groups
Galois Extensions
Decomposition of Primes
Quadratic Extensions
The Cube Roots of Two
Decomposition Groups and Galois Representations
The Decomposition Group
Galois groups of finite fields
The Exact Sequence
Frobenius Elements
Galois Representations and a Conjecture of Artin
Adelic Viewpoint
Valuations
Valuations
Types of Valuations
Examples of Valuations
Topology and Completeness
Topology
Completeness
-adic Numbers
The Field of
-adic Numbers
The Topology of
(is Weird)
The Local-to-Global Principle of Hasse and Minkowski
Weak Approximation
Adic Numbers: The Finite Residue Field Case
Finite Residue Field Case
Normed Spaces and Tensor Products
Normed Spaces
Tensor Products
Extensions and Normalizations of Valuations
Extensions of Valuations
Extensions of Normalized Valuations
Global Fields and Adeles
Global Fields
Restricted Topological Products
The Adele Ring
Strong Approximation
Ideles and Ideals
The Idele Group
Ideals and Divisors
The Function Field Case
Jacobians of Curves
Exercises
Bibliography
Index
About this document ...
William Stein 2004-05-06
