A Brief Guide To Algebraic Number Theory

Broad graduate-level account of Algebraic Number Theory, first published in 2001, including exercises, by a world-renowned author.

This volume honors Sir Peter Swinnerton-Dyer's mathematical career spanning more than 60 years' of amazing creativity in number theory and algebraic geometry.

Bringing the material up to date to reflect modern applications, this second edition has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. It offers a more complete and involved treatment of Galois theory, a more comprehensive section on Pollard's cubic factoring algorithm, and more detailed explanations of proofs to provide a sound understanding of challenging material. This edition also studies binary quadratic forms and compares the ideal and form class groups. The text includes convenient cross-referencing, a comprehensive index, and numerous exercises and applications.

ALGEBRAIC NUMBER THEORY provides concisely both the fundamental and profound theory, starting from the succinct ideal theory (Chapters 1-3), turning then to valuation theory and local completion field (Chapters 4-5) which is the base of modern approach. After specific discussions on class numbers, units, quadratic and cyclotomic fields, and analytical theory (Chapters 6-8), the important Class Field Theory (Chapter 9) is expounded, and algebraic function field (Chapter 10) is sketched. This book is based on the study and lectures of the author at several universities.

The exposition of the classical theory of algebraic numbers is clear and thorough, and there is a large number of exercises as well as worked out numerical examples. A careful study of this book will provide a solid background to the learning of more recent topics.

Ideal either for classroom use or as exercises for mathematically minded individuals, this text introduces elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields.