The theory of algebraic numbers

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Cover; Title Page; Copyright Page; Dedication; Contents; I. Divisibility; 1. Uniqueness of Factorization; 2. A General Problem; 3. The Gaussian Integers; Problems; II. The Gaussian Primes; 1. Rational and Gaussian Primes; 2. Congruences; 3. Determination of the Gaussian Primes; 4. Fermat's Theorem for Gaussian Primes; Problems; III. Polynomials Over a Field; 1. The Ring of Polynomials; 2. The Eisenstein Irreducibility Criterion; 3. Symmetric Polynomials; Problems; IV. Algebraic Number Fields; 1. Numbers Algebraic Over a Field; 2. Extensions of a Field; 3. Algebraic and Transcendental Numbers. ProblemsV. Bases; 1. Bases and Finite Extensions; 2. Properties of Finite Extensions; 3. Conjugates and Discriminants; 4. The Cyclotomic Field; Problems; VI. Algebraic Integers and Integral Bases; 1. Algebraic Integers; 2. The Integers in a Quadratic Field; 3. Integral Bases; 4. Examples of Integral Bases; Problems; VII. Arithmetic in Algebraic Number Fields; 1. Units and Primes; 2. Units in a Quadratic Field; 3. The Uniqueness of Factorization; 4. Ideals in an Algebraic Number Field; Problems; VIII. The Fundamental Theorem of Ideal Theory; 1. Basic Properties of Ideals. 2. The Classical Proof of the Unique Factorization Theorem3. The Modem Proof; Problems; IX. Consequences of the Fundamental Theorem; 1. The Highest Common Factor of Two Ideals; 2. Unique Factorization of Integers; 3. The Problem of Ramification; 4. Congruences and Norms; 5. Further Properties of Norms; Problems; X. Ideal Classes and Class Numbers; 1. Ideal Classes; 2. Class Numbers; Problems; XI. The Fermat Conjecture; 1. Pythagorean Triples; 2. The Fermat Conjecture; 3. Units in Cyclotomic Fields; 4. Kummer's Theorem; Problems; References; List of Symbols; Index