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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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Request a sales quoteThis 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
- No. of pages: 434
- Language: English
- Edition: 1
- Volume: 20
- Published: May 5, 1986
- Imprint: Academic Press
- eBook ISBN: 9780080873329
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