Algebraic Number Theory, Second Edition

Overview

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 ...

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Overview

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.

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Editorial Reviews

From the Publisher
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

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Product Details

Meet the Author

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).

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Table of Contents

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 Series
Appendix C: The Greek Alphabet
Appendix D: Latin Phrases

Bibliography

Solutions to Odd-Numbered Exercises

Index

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