The Theory of Algebraic Numbers
by Pollard, Harry; Diamond, Harold G.Edition: 3rd
Format: Paperback
Pub. Date: 2010-08-19
Publisher(s): Dover Publications
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Summary
An excellent introduction to the basics of algebraic number theory, this concise, well-written volume examines Gaussian primes; polynomials over a field; algebraic number fields; and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal classes and class numbers; and the Fermat conjecture. 1975 edition. References. List of Symbols. Index.
Table of Contents
| Divisibility | |
| Uniqueness of factorization | |
| A general problem | |
| The Gaussian integers | |
| Problems | |
| The Gaussian Primes | |
| Rational and Gaussian primes | |
| Congruences | |
| Determination of the Gaussian primes | |
| Fermat's theorem for Gaussian primes | |
| Problems | |
| Polynomials over a field | |
| The ring of polynomials | |
| The Eisenstein irreducibility criterion | |
| Symmetric polynomials | |
| Problems | |
| Algebraic Number Fields | |
| Numbers algebraic over a field | |
| Extensions of a field | |
| Algebraic and transcendental numbers | |
| Problems | |
| Bases | |
| Bases and finite extensions | |
| Properties of finite extensions | |
| Conjugates and discriminants | |
| The cyclotomic field | |
| Problems | |
| Algebraic Integers and Integral Bases | |
| Algebraic integers | |
| The integers in a quadratic field | |
| Integral bases | |
| Examples of integral bases | |
| Problems | |
| Arithmetic in Algebraic Number Fields | |
| Units and primes | |
| Units in a quadratic field | |
| The uniqueness of factorization | |
| Ideals in an algebraic number field | |
| Problems | |
| The Fundamental Theorem of Ideal Theory | |
| Basic properties of ideals | |
| The classical proof of the unique factorization theorem | |
| The modern proof | |
| Problems | |
| Consequences of the Fundamental Theorem | |
| The highest common factor of two ideals | |
| Unique factorization of integers | |
| The problem of ramification | |
| Congruences and norms | |
| Further properties of norms | |
| Problems | |
| Ideal Classes and Class Numbers | |
| Ideal classes | |
| Class numbers | |
| Problems | |
| The Fermat Conjecture | |
| Pythagorean triples | |
| The Fermat conjecture | |
| Units in cyclotomic fields | |
| Kummer's theorem | |
| Problems | |
| References | |
| List of symbols | |
| Index | |
| Table of Contents provided by Publisher. All Rights Reserved. |
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