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.

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).

Integral Domains, Ideals, and Unique FactorizationIntegral DomainsFactorization DomainsIdealsNoetherian and Principal Ideal DomainsDedekind DomainsAlgebraic Numbers and Number FieldsQuadratic Fields

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

Class GroupsBinary Quadratic FormsForms and IdealsGeometry of Numbers and the Ideal Class GroupUnits in Number RingsDirichlet’s Unit Theorem

Applications: Equations and Sieves Prime Power Representation Bachet’s Equation The Fermat Equation Factoring The Number Field Sieve

Ideal Decomposition in Number FieldsInertia, Ramification, and Splitting of Prime IdealsThe Different and DiscriminantRamificationGalois Theory and DecompositionKummer Extensions and Class-Field TheoryThe Kronecker–Weber TheoremAn Application—Primality Testing

Reciprocity LawsCubic ReciprocityThe Biquadratic Reciprocity LawThe Stickelberger RelationThe Eisenstein Reciprocity Law

Appendix A: Abstract AlgebraAppendix B: Sequences and SeriesAppendix C: The Greek AlphabetAppendix D: Latin Phrases

Bibliography
Solutions to Odd-Numbered Exercises
Index