Number Theory

1st Edition, Volume 20 - May 5, 1986
Editors: Z.I. Borevich, I.R. Shafarevich
Language: English
eBook ISBN:
9 7 8 - 0 - 0 8 - 0 8 7 3 3 2 - 9

This book is written for the student in mathematics. Its goal is to give a view of the theory of numbers, of the problems with which this theory deals, and of the methods that ar… Read more

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This book is written for the student in mathematics. Its goal is to give a view of the theory of numbers, of the problems with which this theory deals, and of the methods that are used.

We have avoided that style which gives a systematic development of the apparatus and have used instead a freer style, in which the problems and the methods of solution are closely interwoven. We start from concrete problems in number theory. General theories arise as tools for solving these problems. As a rule, these theories are developed sufficiently far so that the reader can see for himself their strength and beauty, and so that he learns to apply them.

Most of the questions that are examined in this book are connected with the theory of diophantine equations - that is, with the theory of the solutions in integers of equations in several variables. However, we also consider questions of other types; for example, we derive the theorem of Dirichlet on prime numbers in arithmetic progressions and investigate the growth of the number of solutions of congruences.

Pure and Applied Mathematics

Translator’s Preface

Foreword

Chapter 1: Congruences

Problems

1. Congruences with Prime Modulus

Problems

2. Trigonometric Sums

Problems

3. p-Adic Numbers

Problems

4. An Axiomatic Characterization of the Field of p-Adic Numbers

Problems

5. Congruences and p-Adic Integers

Problems

6. Quadratic Forms with p-Adic Coefficients

Problems

7. Rational Quadratic Forms

Problems

Chapter 2: Representation of Numbers by Decomposable Forms

1. Decomposable Forms

Problems

2. Full Modules and Their Rings of Coefficients

Problems

3. Geometric Methods

Problems

4. The Group of Units

Problems

5. The Solution of the Problem of the Representation of Rational Numbers by Full Decomposable Forms

Problems

6. Classes of Modules

7. Representation of Numbers by Binary Quadratic Forms

7.3. Units

7.4. Modules

7.5. The Correspondence between Modules and Forms

7.6. The Representation of Numbers by Binary Forms and Similarity of Modules

7.7. Similarity of Modules in Imaginary Quadratic Fields

Theorem 8.

Definition.

Theorem 9.

Remark.

Example 1.

Example 2.

Example 3.

Problems

Chapter 3: The Theory of Divisibility

1. Some Special Cases of Fermat’s Theorem

Problems

2. Decomposition into Factors

Problems

3. Divisors

Problems

4. Valuations

Problems

5. Theories of Divisors for Finite Extensions

Problems

6. Dedekind Rings

Problems

7. Divisors in Algebraic Number Fields

Problems

8. Quadratic Fields

Problems

Chapter 4: Local Methods

1. Fields Complete with Respect to a Valuation

Problems

2. Finite Extensions of Fields with Valuations

3. Factorization of Polynomials in a Field Complete with Respect to a Valuation

Problems

4. Metrics on Algebraic Number Fields

Problems

5. Analytic Functions in Complete Fields

Problems

6. Skolem’s Method

Problems

7. Local Analytic Manifolds

Chapter 5: Analytic Methods

1. Analytic Formulas for the Number of Divisor Classes

Problems

2. The Number of Divisor Classes of Cyclotomic Fields

Problems

3. Dirichlet’s Theorem on Prime Numbers in Arithmetic Progressions

Problems

4. The Number of Divisor Classes of Quadratic Fields

Problems

5. The Number of Divisor Classes of Prime Cyclotomic Fields

Problems

6. A Criterion for Regularity

Problems

7. The Second Case of Fermat’s Theorem for Regular Exponents

Problems

8. Bernoulli Numbers

Problems

Algebraic Supplement

1. Quadratic Forms over Arbitrary Fields of Characteristic ≠ 2

Problems

2. Algebraic Extensions

Problems

3. Finite Fields

Problems

4. Some Results on Commutative Rings

Problems

5. Characters

Problems

Tables

Index