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Basic Commutative Algebra
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ALGEBRAIC NUMBER THEORY, A
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Some applications of algebraic
Contents
Algebraic Number Fields
Subsections
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
William Stein 2008-10-03
-series and a Conjecture of
Artin
-functions
