442 Pages
    by Chapman & Hall

    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
    Integral Domains
    Factorization Domains
    Noetherian and Principal Ideal Domains
    Dedekind Domains
    Algebraic Numbers and Number Fields
    Quadratic Fields

    Field Extensions
    Automorphisms, Fixed Points, and Galois Groups
    Norms and Traces
    Integral Bases and Discriminants
    Norms of Ideals

    Class Groups
    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
    Bachet’s Equation
    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

    Reciprocity Laws
    Cubic Reciprocity
    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



    Richard A. Mollin is a professor in the Department of Mathematics and Statistics at the University of Calgary. In the past twenty-five years, Dr. Mollin has founded the Canadian Number Theory Association and has been awarded six Killam Resident Fellowships. He has written more than 200 publications, including Advanced Number Theory with Applications (CRC Press, August 2009), Fundamental Number Theory with Applications, Second Edition (CRC Press, February 2008), An Introduction to Cryptography, Second Edition (CRC Press, September 2006), Codes: The Guide to Secrecy from Ancient to Modern Times (CRC Press, May 2005), and RSA and Public-Key Cryptography (CRC Press, November 2002).

    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