Number Theory (eBook)

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1986 | 1. Auflage
434 Seiten
Elsevier Science (Verlag)
978-0-08-087332-9 (ISBN)

Infos

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.

Inhalt

Front Cover 1
Number Theory 4
Copyright Page 5
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’s 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’s 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’s 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’s 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

Produkt-Details

• Erscheint lt. Verlag: 5.5.1986
• Mitarbeit: Herausgeber (Serie): Z.I. Borevich, I.R. Shafarevich
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Technik
• ISBN-10: 0-08-087332-4 / 0080873324
• ISBN-13: 978-0-08-087332-9 / 9780080873329