Number Theory

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.

1;Front Cover;1 2;Number Theory;4 3;Copyright Page;5 4;Contents;10 5;Translator's Preface;6 6;Foreword;8 7;Chapter 1. Congruences;14 7.1;1. Congruences with Prime Modulus;16 7.2;2. Trigonometric Sums;22 7.3;3. p-Adic Numbers;31 7.4;4. An Axiomatic Characterization of the Field of p-adic Numbers;45 7.5;5. Congruences and p-adic Integers;53 7.6;6. Quadratic Forms with p-adic Coefficients;60 7.7;7. Rational Quadratic Forms;74 8;Chapter 2. Representation of Numbers by Decomposable Forms;88 8.1;1. Decomposable Forms;90 8.2;2. Full Modules and Their Rings of Coefficients;96 8.3;3. Geometric Methods;107 8.4;4. The Groups of Units;120 8.5;5. The Solution of the Problem of the Representation of Rational Numbers by Full Decomposable Forms;129 8.6;6. Classes of Modules;136 8.7;7. Representation of Numbers by Binary Quadratic Forms;142 9;Chapter 3. The Theory of Divisibility;168 9.1;1. Some Special Cases of Fermats Theorem;169 9.2;2. Decomposition into Factors;177 9.3;3. Divisors;183 9.4;4. Valuations;193 9.5;5. Theories of Divisors for Finite Extensions;206 9.6;6. Dedekind Rings;220 9.7;7. Divisors in Algebraic Number Fields;229 9.8;8. Quadratic Fields;247 10;Chapter 4. Local Methods;264 10.1;1. Fields Complete with Respect to a Valuation;266 10.2;2. Finite Extensions of Fields with Valuations;280 10.3;3. Factorization of Polynomials in a Field Complete with Respect to a Valuation;285 10.4;4. Metrics on Algebraic Number Fields;290 10.5;5. Analytic Functions in Complete Fields;295 10.6;6. Skolems Method;303 10.7;7. Local Analytic Manifolds;315 11;Chapter 5. Analytic Methods;322 11.1;1. Analytic Formulas for the Number of Divisor Classes;322 11.2;2. The Number of Divisor Classes of Cyclotomic Fields;338 11.3;3. Dirichlets Theorem on Prime Numbers in Arithmetic Progressions;351 11.4;4. The Number of Divisor Classes of Quadratic Fields;355 11.5;5. The Number of Divisor Classes of Prime Cyclotomic Fields;368 11.6;6. A Criterion for Regularity;380 11.7;7. The Second Case of Fermats
Theorem for Regular Exponents;391 11.8;8. Bernoulli Numbers;395 12;Algebraic Supplement;403 12.1;1. Quadratic Forms over Arbitrary Fields of Characteristic # 2;403 12.2;2. Algebraic Extensions;409 12.3;3. Finite Fields;418 12.4;4. Some Results on Commutative Rings;423 12.5;5. Characters;428 13;Tables;435 14;Subject Index;446