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Contents
Contents
A
LGEBRAIC
N
UMBER
T
HEORY,
A
C
OMPUTATIONAL
A
PPROACH
William Stein
Contents
Preface
Introduction
Mathematical background
What is algebraic number theory?
Topics in this book
Some applications of algebraic number theory
Algebraic Number Fields
Basic Commutative Algebra
Finitely Generated Abelian Groups
Noetherian Rings and Modules
The Ring
is noetherian
Rings of Algebraic Integers
Norms and Traces
Recognizing Algebraic Numbers using Lattice Basis Reduction (LLL)
LLL Reduced Basis
What LLL really means
Applying LLL
Dedekind Domains and Unique Factorization of Ideals
Dedekind Domains
Factoring Primes
The Problem
Geometric Intuition
Examples
A Method for Factoring Primes that Often Works
A General Method
Inessential Discriminant Divisors
Remarks on Ideal Factorization in General
Finding a
-Maximal Order
General Factorization Algorithm of Buchman-Lenstra
The Chinese Remainder Theorem
The Chinese Remainder Theorem
CRT in the Integers
CRT in General
Structural Applications of the CRT
Computing Using the CRT
.
PARI
Discrimants and Norms
Viewing
as a Lattice in a Real Vector Space
The Volume of
Discriminants
Norms of Ideals
Finiteness of the Class Group
The Class Group
Class Number 1
More About Computing Class Groups
Dirichlet's Unit Theorem
The Group of Units
Examples with MAGMA
Pell's Equation
Examples with Various Signatures
Decomposition and Inertia Groups
Galois Extensions
Decomposition of Primes:
Quadratic Extensions
The Cube Root of Two
The Decomposition Group
Galois groups of finite fields
The Exact Sequence
Frobenius Elements
Galois Representations,
-series and a Conjecture of Artin
Elliptic Curves, Galois Representations, and
-functions
Groups Attached to Elliptic Curves
Abelian Groups Attached to Elliptic Curves
A Formula for Adding Points
Other Groups
Galois Representations Attached to Elliptic Curves
Modularity of Elliptic Curves over
Galois Cohomology
Group Cohomology
Group Rings
Modules and Group Cohomology
Example Application of the Theorem
Inflation and Restriction
Galois Cohomology
The Weak Mordell-Weil Theorem
Kummer Theory of Number Fields
Proof of the Weak Mordell-Weil Theorem
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
About this document ...
William Stein 2008-10-03
-functions
