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- Explains how to solve Diophantine equations
- Describes applications of factoring, the number field sieve, primality testing, and cryptography
- Contains more than forty mini-biographies of notable mathematicians in algebraic number theory
- Reviews all of the requisite background material in the appendices
- Provides convenient cross-referencing and a comprehensive index of over 2,000 entries
- Includes over 300 exercises that test and challenge readers as well as illustrate concepts, with solutions to odd-numbered exercises at the back of the book

*Solutions manual available for qualifying instructors *

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

Ideals

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

Factoring

The Number Field Sieve

**Ideal Decomposition in Number Fields**

Inertia, Ramification, and Splitting of Prime Ideals

The Different and Discriminant

Ramification

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 SeriesAppendix C: The Greek AlphabetAppendix D: Latin Phrases**

Bibliography

**Solutions to Odd-Numbered Exercises **

**Index**

**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