Introductory Algebraic Number Theory

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 flavour. Prerequisites are kept to a minimum, and numerous examples illustrating the material occur throughout the text. References to suggested reading 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.

Table of 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.

Review :
'The overall presentation makes the book suitable for a course for advanced undergraduate students.' Zentralblatt MATH 'Learning algebraic number theory is about the least abstract way to learn about important aspects of commutative ring theory, as well as being beautiful in its own right too. This text is ideally suited to the learner of both of these, with clear writing, a plentiful supply of examples and exercises, and a good range of 'suggested reading'. ... I look forward to reading and learning from this book in greater detail. The features which make it attractive are worth listing: the intrinsic fascination of the results; the balance between clearn theory and dirty calculation (the latter essential for developing familiarity with the local terrain, the former for appreciating an arial view of the whole route); the balance between calculation dependent upon the depth of theory and those details dependent on simple algebraic and trigonometric identities and results from elementary number theory; a very full quota of exercises and further reading.' The Mathematical Gazette 'This is a very good textbook on algebraic number theory for beginners. ... Its most appealing feature is the very large number of examples it contains. Their abundance provides a lot of hands-on experience and has the power to transform the reader's understanding of basic notions into active knowledge.' Monatshefte fur Mathematik 'This book provides a nice introduction to classical parts of algebraic number theory... The text is written in a lively style and can be read without any prerequisites. Therefore the book is very suitable for graduate students starting mathematics courses or mathematicians interested in introductory reading in algebraic number theory. The book presents a welcome addition to the existing literature.' EMS Newsletter