Number theory

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English Pages [448] Year 1986

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Number theory

Table of contents :
Number Theory
Copyright Page
Contents
Translator's Preface
Foreword
Chapter 1. Congruences
1. Congruences with Prime Modulus
2. Trigonometric Sums
3. p-Adic Numbers
4. An Axiomatic Characterization of the Field of p-adic Numbers
5. Congruences and p-adic Integers
6. Quadratic Forms with p-adic Coefficients
7. Rational Quadratic Forms
Chapter 2. Representation of Numbers by Decomposable Forms
1. Decomposable Forms
2. Full Modules and Their Rings of Coefficients
3. Geometric Methods
4. The Groups of Units
5. The Solution of the Problem of the Representation of Rational Numbers by Full Decomposable Forms
6. Classes of Modules
7. Representation of Numbers by Binary Quadratic Forms
Chapter 3. The Theory of Divisibility
1. Some Special Cases of Fermats Theorem
2. Decomposition into Factors
3. Divisors
4. Valuations
5. Theories of Divisors for Finite Extensions
6. Dedekind Rings
7. Divisors in Algebraic Number Fields
8. Quadratic Fields
Chapter 4. Local Methods
1. Fields Complete with Respect to a Valuation
2. Finite Extensions of Fields with Valuations
3. Factorization of Polynomials in a Field Complete with Respect to a Valuation
4. Metrics on Algebraic Number Fields
5. Analytic Functions in Complete Fields
6. Skolems Method
7. Local Analytic Manifolds
Chapter 5. Analytic Methods
1. Analytic Formulas for the Number of Divisor Classes
2. The Number of Divisor Classes of Cyclotomic Fields
3. Dirichlets Theorem on Prime Numbers in Arithmetic Progressions
4. The Number of Divisor Classes of Quadratic Fields
5. The Number of Divisor Classes of Prime Cyclotomic Fields
6. A Criterion for Regularity
7. The Second Case of Fermats Theorem for Regular Exponents
8. Bernoulli Numbers
Algebraic Supplement
1. Quadratic Forms over Arbitrary Fields of Characteristic 2
2. Algebraic Extensions
3. Finite Fields
4. Some Results on Commutative Rings
5. Characters
Tables
Subject Index

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