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Pure and Applied Mathematics
Chapter 1: Congruences
1. Congruences with Prime Modulus
2. Trigonometric Sums
3. p-Adic Numbers
4. An Axiomatic Characterization of the Field of p-Adic Numbers
5. Congruences and p-Adic Integers
6. Quadratic Forms with p-Adic Coefficients
7. Rational Quadratic Forms
Chapter 2: Representation of Numbers by Decomposable Forms
1. Decomposable Forms
2. Full Modules and Their Rings of Coefficients
3. Geometric Methods
4. The Group of Units
5. The Solution of the Problem of the Representation of Rational Numbers by Full Decomposable Forms
6. Classes of Modules
7. Representation of Numbers by Binary Quadratic Forms
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
Chapter 3: The Theory of Divisibility
1. Some Special Cases of Fermat’s Theorem
2. Decomposition into Factors
5. Theories of Divisors for Finite Extensions
6. Dedekind Rings
7. Divisors in Algebraic Number Fields
8. Quadratic Fields
Chapter 4: Local Methods
1. Fields Complete with Respect to a Valuation
2. Finite Extensions of Fields with Valuations
3. Factorization of Polynomials in a Field Complete with Respect to a Valuation
4. Metrics on Algebraic Number Fields
5. Analytic Functions in Complete Fields
6. Skolem’s Method
7. Local Analytic Manifolds
Chapter 5: Analytic Methods
1. Analytic Formulas for the Number of Divisor Classes
2. The Number of Divisor Classes of Cyclotomic Fields
3. Dirichlet’s Theorem on Prime Numbers in Arithmetic Progressions
4. The Number of Divisor Classes of Quadratic Fields
5. The Number of Divisor Classes of Prime Cyclotomic Fields
6. A Criterion for Regularity
7. The Second Case of Fermat’s Theorem for Regular Exponents
8. Bernoulli Numbers
1. Quadratic Forms over Arbitrary Fields of Characteristic ≠ 2
2. Algebraic Extensions
3. Finite Fields
4. Some Results on Commutative Rings
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.
- No. of pages:
- © Academic Press 1986
- 5th May 1986
- Academic Press
- eBook ISBN:
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