Number Theory

Number Theory Copyright Page Contents Translator's Preface Foreword 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 Groups 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 Chapter 3. The Theory of Divisibility 1. Some Special Cases of Fermat’s Theorem 2. Decomposition into Factors 3. Divisors 4. Valuations 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 Algebraic Supplement 1. Quadratic Forms over Arbitrary Fields of Characteristic 2 2. Algebraic Extensions 3. Finite Fields 4. Some Results on Commutative Rings 5. Characters Tables Subject Index