简介
Modern number theory, according to Hecke, dates from Gauss's quadratic reciprocity law. The various extensions of this law and the generalizations of the domains of study for number theory have led to a rich network of ideas, which has had effects throughout mathematics, in particular in algebra. This volume of the Encyclopaedia presents the main structures and results of algebraic number theory with emphasis on algebraic number fields and class field theory. Koch has written for the non-specialist. He assumes that the reader has a general understanding of modern algebra and elementary number theory. Mostly only the general properties of algebraic number fields and related structures are included. Special results appear only as examples which illustrate general features of the theory. A part of algebraic number theory serves as a basic science for other parts of mathematics, such as arithmetic algebraic geometry and the theory of modular forms. For this reason, the chapters on basic number theory, class field theory and Galois cohomology contain more detail than the others. This book is suitable for graduate students and research mathematicians who wish to become acquainted with the main ideas and methods of algebraic number theory.
目录
Contents 10
Translator's Preface 6
Foreword 8
Chapter 1. Congruences 14
1. Congruences with Prime Modulus 16
2. Trigonometric Sums 22
3. p-Adic Numbers 31
4. An Axiomatic Characterization of the Field of p-adic Numbers 45
5. Congruences and p-adic Integers 53
6. Quadratic Forms with p-adic Coefficients 60
7. Rational Quadratic Forms 74
Chapter 2. Representation of Numbers by Decomposable Forms 88
1. Decomposable Forms 90
2. Full Modules and Their Rings of Coefficients 96
3. Geometric Methods 107
4. The Groups of Units 120
5. The Solution of the Problem of the Representation of Rational Numbers by Full Decomposable Forms 129
6. Classes of Modules 136
7. Representation of Numbers by Binary Quadratic Forms 142
Chapter 3. The Theory of Divisibility 168
1. Some Special Cases of Fermat\u2019s Theorem 169
2. Decomposition into Factors 177
3. Divisors 183
4. Valuations 193
5. Theories of Divisors for Finite Extensions 206
6. Dedekind Rings 220
7. Divisors in Algebraic Number Fields 229
8. Quadratic Fields 247
Chapter 4. Local Methods 264
1. Fields Complete with Respect to a Valuation 266
2. Finite Extensions of Fields with Valuations 280
3. Factorization of Polynomials in a Field Complete with Respect to a Valuation 285
4. Metrics on Algebraic Number Fields 290
5. Analytic Functions in Complete Fields 295
6. Skolem\u2019s Method 303
7. Local Analytic Manifolds 315
Chapter 5. Analytic Methods 322
1. Analytic Formulas for the Number of Divisor Classes 322
2. The Number of Divisor Classes of Cyclotomic Fields 338
3. Dirichlet\u2019s Theorem on Prime Numbers in Arithmetic Progressions 351
4. The Number of Divisor Classes of Quadratic Fields 355
5. The Number of Divisor Classes of Prime Cyclotomic Fields 368
6. A Criterion for Regularity 380
7. The Second Case of Fermat\u2019s Theorem for Regular Exponents 391
8. Bernoulli Numbers 395
Algebraic Supplement 403
1. Quadratic Forms over Arbitrary Fields of Characteristic # 2 403
2. Algebraic Extensions 409
3. Finite Fields 418
4. Some Results on Commutative Rings 423
5. Characters 428
Tables 435
Subject Index 446
Translator's Preface 6
Foreword 8
Chapter 1. Congruences 14
1. Congruences with Prime Modulus 16
2. Trigonometric Sums 22
3. p-Adic Numbers 31
4. An Axiomatic Characterization of the Field of p-adic Numbers 45
5. Congruences and p-adic Integers 53
6. Quadratic Forms with p-adic Coefficients 60
7. Rational Quadratic Forms 74
Chapter 2. Representation of Numbers by Decomposable Forms 88
1. Decomposable Forms 90
2. Full Modules and Their Rings of Coefficients 96
3. Geometric Methods 107
4. The Groups of Units 120
5. The Solution of the Problem of the Representation of Rational Numbers by Full Decomposable Forms 129
6. Classes of Modules 136
7. Representation of Numbers by Binary Quadratic Forms 142
Chapter 3. The Theory of Divisibility 168
1. Some Special Cases of Fermat\u2019s Theorem 169
2. Decomposition into Factors 177
3. Divisors 183
4. Valuations 193
5. Theories of Divisors for Finite Extensions 206
6. Dedekind Rings 220
7. Divisors in Algebraic Number Fields 229
8. Quadratic Fields 247
Chapter 4. Local Methods 264
1. Fields Complete with Respect to a Valuation 266
2. Finite Extensions of Fields with Valuations 280
3. Factorization of Polynomials in a Field Complete with Respect to a Valuation 285
4. Metrics on Algebraic Number Fields 290
5. Analytic Functions in Complete Fields 295
6. Skolem\u2019s Method 303
7. Local Analytic Manifolds 315
Chapter 5. Analytic Methods 322
1. Analytic Formulas for the Number of Divisor Classes 322
2. The Number of Divisor Classes of Cyclotomic Fields 338
3. Dirichlet\u2019s Theorem on Prime Numbers in Arithmetic Progressions 351
4. The Number of Divisor Classes of Quadratic Fields 355
5. The Number of Divisor Classes of Prime Cyclotomic Fields 368
6. A Criterion for Regularity 380
7. The Second Case of Fermat\u2019s Theorem for Regular Exponents 391
8. Bernoulli Numbers 395
Algebraic Supplement 403
1. Quadratic Forms over Arbitrary Fields of Characteristic # 2 403
2. Algebraic Extensions 409
3. Finite Fields 418
4. Some Results on Commutative Rings 423
5. Characters 428
Tables 435
Subject Index 446
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