Introduction to the Theory of Algebraic Numbers and Fuctions (eBook)

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