Algebraic Number Theory
Bringing the material up to date to reflect modern applications, Algebraic Number Theory, Second Edition has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. This edition focuses on integral domains, ideals, and unique factorization in the first chapter; field extensions in the second chapter; and class groups in the third chapter. Applications are now collected in chapter four and at the end of chapter five, where primality testing is highlighted as an application of the Kronecker–Weber theorem. In chapter five, the sections on ideal decomposition in number fields have been more evenly distributed. The final chapter continues to cover reciprocity laws.
New to the Second Edition
- Reorganization of all chapters
- More complete and involved treatment of Galois theory
- A study of binary quadratic forms and a comparison of the ideal and form class groups
- More comprehensive section on Pollard’s cubic factoring algorithm
- More detailed explanations of proofs, with less reliance on exercises, to provide a sound understanding of challenging material
The book includes mini-biographies of notable mathematicians, convenient cross-referencing, a comprehensive index, and numerous exercises. The appendices present an overview of all the concepts used in the main text, an overview of sequences and series, the Greek alphabet with English transliteration, and a table of Latin phrases and their English equivalents.
Suitable for a one-semester course, this accessible, self-contained text offers broad, in-depth coverage of numerous applications. Readers are lead at a measured pace through the topics to enable a clear understanding of the pinnacles of algebraic number theory.
Integral Domains, Ideals, and Unique Factorization
Noetherian and Principal Ideal Domains
Algebraic Numbers and Number Fields
Automorphisms, Fixed Points, and Galois Groups
Norms and Traces
Integral Bases and Discriminants
Norms of Ideals
Binary Quadratic Forms
Forms and Ideals
Geometry of Numbers and the Ideal Class Group
Units in Number Rings
Dirichlet’s Unit Theorem
Applications: Equations and Sieves
Prime Power Representation
The Fermat Equation
The Number Field Sieve
Ideal Decomposition in Number Fields
Inertia, Ramification, and Splitting of Prime Ideals
The Different and Discriminant
Galois Theory and Decomposition
Kummer Extensions and Class-Field Theory
The Kronecker–Weber Theorem
An Application—Primality Testing
The Biquadratic Reciprocity Law
The Stickelberger Relation
The Eisenstein Reciprocity Law
Appendix A: Abstract Algebra
Appendix B: Sequences and Series
Appendix C: The Greek Alphabet
Appendix D: Latin Phrases
Solutions to Odd-Numbered Exercises
This is an introductory text in algebraic number theory that has good coverage … . This second edition is completely reorganized and rewritten from the first edition. … Very Good Features: (1) The applications are not limited to Diophantine equations, as in many books, but cover a wide range, including factorization into primes, primality testing, and the higher reciprocity laws. (2) The book has a large number of mini-biographies of the number theorists whose work is being discussed. …
—MAA Reviews, April 2011
This book is in the MAA's basic library list. The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.
Praise for the First Edition
This is a remarkable book that will be a valuable reference for many people, including me. The book shows great care in preparation, and the ample details and motivation will be appreciated by lots of students. The solid punches at the end of each chapter will be appreciated by everybody. It deserves success with many adoptions as a text.
—Irving Kaplansky, Mathematical Sciences Research Institute, Berkeley, California, USA
An extremely well-written and clear presentation of algebraic number theory suitable for beginning graduate students. The many exercises, applications, and references are a very valuable feature of the book.
—Kenneth Williams, Carleton University, Ottawa, Ontario, Canada
This is a unique book that will be influential.
—John Brillhart, University of Arizona, Tucson, USA