Introduction to the Theory of Algebraic Numbers and Functions

Introduction to The Theory of Algebraic Numbers and Functions......Page 4
Copyright Page......Page 5
Contents......Page 10
Preface to The English Edition......Page 6
Preface to The German Edition......Page 8
1. The Subject......Page 18
2. The Method......Page 19
Table of Several Abbreviations and Symbols......Page 20
1. Modules in Principal Ideal Domains......Page 22
2. Systems of Linear Inequalities......Page 31
3. Linear Divisors......Page 37
4. Traces, Norms, and Discriminants......Page 44
1. The Symplectic Group......Page 49
2. Theta Functions for Quadratic Forms......Page 61
1. Ideals......Page 70
2. Local Rings......Page 80
3. Ideals in Different Fields; the Norm......Page 87
4. The Complement, Different, and Discriminant......Page 90
5. Divisors......Page 96
6. Decomposition of Prime Ideals in Galois Extensions......Page 107
1. The Finiteness Theorems......Page 115
2. Quadratic Number Fields and Cyclotomic Fields......Page 121
1. Power Series Expansions of Algebraic Functions......Page 127
2. Algebraic Function Fields......Page 137
3. The Riemann-Roch Theorem......Page 149
4. Differentials......Page 160
5. Differentials and Principal Part Systems......Page 175
6. Reduction of a Function Field with Respect to a Prime Ideal of the Constant Field......Page 188
1. Riemann Surfaces......Page 202
2. Fields of Elliptic Functions......Page 207
3. The Group of Divisor Classes of Degree 0......Page 221
4. Modular Functions......Page 231
1. The Correspondences......Page 250
2. Representations of Correspondences in the Space of Differentials......Page 269
3. Modular Functions......Page 283
4. Castelnuovo's Inequality......Page 298
5. Applications in Number Theory......Page 316
6. Elliptic Function Fields......Page 332
Author Index......Page 338
Subject Index......Page 339