Theory of Algebraic Numbers (eBook)

Chapter I. Divisibility 1. Uniqueness of factorization 2. A general problem 3. The Gaussian integers ProblemsChapter II. The Gaussian Primes 1. Rational and Gaussian primes 2. Congruences 3. Determination of the Gaussian primes 4. Fermat's theorem for Gaussian primes ProblemsChapter III. Polynomials over a field 1. The ring of polynomials 2. The Eisenstein irreducibility criterion 3. Symmetric polynomials ProblemsChapter IV. Algebraic Number Fields 1. Numbers algebraic over a field 2. Extensions of a field 3. Algebraic and transcendental numbers ProblemsChapter V. Bases 1. Bases and finite extensions 2. Properties of finite extensions 3. Conjugates and discriminants 4. The cyclotomic field ProblemsChapter VI. Algebraic Integers and Integral Bases 1. Algebraic integers 2. The integers in a quadratic field 3. Integral bases 4. Examples of integral bases ProblemsChapter 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 ProblemsChapter VIII. The Fundamental Theorem of Ideal Theory 1. Basic properties of ideals 2. The classical proof of the unique factorization theorem 3. The modern proof ProblemsChapter 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 ProblemsChapter X. Ideal Classes and Class Numbers 1. Ideal classes 2. Class numbers ProblemsChapter 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