Introductory Algebraic Number Theory

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by Saban Alaca and Kenneth S. Williams

Cambridge University Press

Due/Published January 2004, 448 pages, paper

ISBN 0521540119

Algebraic number theory is a subject which came into being through the attempts of mathematicians to try to prove Fermat's last theorem and which now has a wealth of applications to diophantine equations, cryptography, factoring, primality testing and public-key cryptosystems. This book provides an introduction to the subject suitable for senior undergraduates and beginning graduate students in mathematics. The material is presented in a straightforward, clear and elementary fashion, and the approach is hands on, with an explicit computational flavor. Prerequisites are kept to a minimum, and numerous examples illustrating the material occur throughout the text. References to suggested readings and to the biographies of mathematicians who have contributed to the development of algebraic number theory are given at the end of each chapter. There are over 320 exercises, an extensive index, and helpful location guides to theorems and lemmas in the text. 16 tables 320 exercises

 Contents

Introduction
1. Integral domains
2. Euclidean domains
3. Noetherian domains
4. Elements integral over a domain
5. Algebraic extensions of a field
6. Algebraic number fields
7. Integral bases
8. Dedekind domains
9. Norms of ideals
10. Decomposing primes in a number field
11. Units in real quadratic fields
12. The ideal class group
13. Dirichlet's unit theorem
14. Applications to diophantine equations.