Algebraic number theory, a computational approach

✍ Scribed by Stein W.A.

• Year: 2012
• Tongue: English
• Leaves: 215
• Edition: free web version
• Category: Library

✦ Table of Contents

What is algebraic number theory?......Page 9
Some applications of algebraic number theory......Page 10
I Algebraic Number Fields......Page 13
Finitely Generated Abelian Groups......Page 15
Noetherian Rings and Modules......Page 20
The Ring Z is noetherian......Page 24
Rings of Algebraic Integers......Page 25
Minimal Polynomials......Page 26
Number fields, rings of integers, and orders......Page 30
Norms and Traces......Page 32
Recognizing Algebraic Numbers using Lattice Basis Reduction (LLL)......Page 35
LLL Reduced Basis......Page 36
What LLL really means......Page 38
Applying LLL......Page 39
Dedekind Domains......Page 41
The Problem......Page 49
Geometric Intuition......Page 50
Examples......Page 51
A Method for Factoring Primes that Often Works......Page 52
Inessential Discriminant Divisors......Page 55
Remarks on Ideal Factorization in General......Page 56
General Factorization Algorithm of Buchman-Lenstra......Page 57
CRT in the Integers......Page 61
CRT in General......Page 62
Structural Applications of the CRT......Page 63
Computing Using the CRT......Page 65
Magma......Page 66
PARI......Page 67
Viewing OK as a Lattice in a Real Vector Space......Page 69
A Determinant......Page 70
Discriminants......Page 71
Norms of Ideals......Page 74
The Class Group......Page 77
Class Number 1......Page 83
More About Computing Class Groups......Page 84
The Group of Units......Page 87
Pell's Equation......Page 93
Examples with Various Signatures......Page 94
Galois Extensions......Page 99
Decomposition of Primes: efg=n......Page 101
Quadratic Extensions......Page 102
The Cube Root of Two......Page 103
The Decomposition Group......Page 104
Galois groups of finite fields......Page 105
The Exact Sequence......Page 106
Frobenius Elements......Page 107
Galois Representations, L-series and a Conjecture of Artin......Page 108
Groups Attached to Elliptic Curves......Page 111
Abelian Groups Attached to Elliptic Curves......Page 112
Other Groups......Page 115
Galois Representations Attached to Elliptic Curves......Page 116
Modularity of Elliptic Curves over Q......Page 118
Modules and Group Cohomology......Page 121
Example Application of the Theorem......Page 123
Inflation and Restriction......Page 124
Galois Cohomology......Page 125
Kummer Theory of Number Fields......Page 127
Proof of the Weak Mordell-Weil Theorem......Page 129
II Adelic Viewpoint......Page 133
Valuations......Page 135
Types of Valuations......Page 137
Examples of Valuations......Page 141
Topology......Page 145
Completeness......Page 147
p-adic Numbers......Page 148
The Field of p-adic Numbers......Page 151
The Topology of QN (is Weird)......Page 152
Weak Approximation......Page 153
Finite Residue Field Case......Page 157
Normed Spaces......Page 165
Tensor Products......Page 167
Extensions of Valuations......Page 173
Extensions of Normalized Valuations......Page 178
Global Fields......Page 181
Restricted Topological Products......Page 185
The Adele Ring......Page 186
Strong Approximation......Page 190
The Idele Group......Page 195
Ideals and Divisors......Page 199
Jacobians of Curves......Page 200
Exercises......Page 201