Number Theory

Number Theory

1st Edition - May 5, 1986

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  • Editors: Z.I. Borevich, I.R. Shafarevich
  • eBook ISBN: 9780080873329

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Description

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.

Table of Contents

  • 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

Product details

  • No. of pages: 434
  • Language: English
  • Copyright: © Academic Press 1986
  • Published: May 5, 1986
  • Imprint: Academic Press
  • eBook ISBN: 9780080873329

About the Series Editors

Z.I. Borevich

I.R. Shafarevich

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