4th Edition

Algebraic Number Theory and Fermat's Last Theorem

ISBN 9781498738392
Published October 13, 2015 by Chapman and Hall/CRC
322 Pages - 21 B/W Illustrations

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Book Description

Updated to reflect current research, Algebraic Number Theory and Fermat’s Last Theorem, Fourth Edition introduces fundamental ideas of algebraic numbers and explores one of the most intriguing stories in the history of mathematics—the quest for a proof of Fermat’s Last Theorem. The authors use this celebrated theorem to motivate a general study of the theory of algebraic numbers from a relatively concrete point of view. Students will see how Wiles’s proof of Fermat’s Last Theorem opened many new areas for future work.

New to the Fourth Edition

  • Provides up-to-date information on unique prime factorization for real quadratic number fields, especially Harper’s proof that Z(√14) is Euclidean
  • Presents an important new result: Mihăilescu’s proof of the Catalan conjecture of 1844
  • Revises and expands one chapter into two, covering classical ideas about modular functions and highlighting the new ideas of Frey, Wiles, and others that led to the long-sought proof of Fermat’s Last Theorem
  • Improves and updates the index, figures, bibliography, further reading list, and historical remarks

Written by preeminent mathematicians Ian Stewart and David Tall, this text continues to teach students how to extend properties of natural numbers to more general number structures, including algebraic number fields and their rings of algebraic integers. It also explains how basic notions from the theory of algebraic numbers can be used to solve problems in number theory.

Table of Contents

Algebraic Methods
Algebraic Background
Rings and Fields
Factorization of Polynomials
Field Extensions
Symmetric Polynomials
Free Abelian Groups

Algebraic Numbers
Algebraic Numbers
Conjugates and Discriminants
Algebraic Integers
Integral Bases
Norms and Traces
Rings of Integers

Quadratic and Cyclotomic Fields
Quadratic Fields
Cyclotomic Fields

Factorization into Irreducibles
Historical Background
Trivial Factorizations
Factorization into Irreducibles
Examples of Non-Unique Factorization into Irreducibles
Prime Factorization
Euclidean Domains
Euclidean Quadratic Fields
Consequences of Unique Factorization
The Ramanujan–Nagell Theorem

Historical Background
Prime Factorization of Ideals
The Norm of an Ideal
Nonunique Factorization in Cyclotomic Fields

Geometric Methods
The Quotient Torus

Minkowski's Theorem
Minkowski's Theorem
The Two-Squares Theorem
The Four-Squares Theorem

Geometric Representation of Algebraic Numbers
The Space Lst

Class-Group and Class-Number
The Class-Group
An Existence Theorem
Finiteness of the Class-Group
How to Make an Ideal Principal
Unique Factorization of Elements in an Extension Ring

Number-Theoretic Applications
Computational Methods
Factorization of a Rational Prime
Minkowski Constants
Some Class-Number Calculations
Table of Class-Numbers

Kummer's Special Case of Fermat's Last Theorem
Some History
Elementary Considerations
Kummer's Lemma
Kummer's Theorem
Regular Primes

The Path to the Final Breakthrough
The Wolfskehl Prize
Other Directions
Modular Functions and Elliptic Curves
The Taniyama–Shimura–Weil Conjecture
Frey's Elliptic Equation
The Amateur Who Became a Model Professional
Technical Hitch
Flash of Inspiration

Elliptic Curves
Review of Conics
Projective Space
Rational Conics and the Pythagorean Equation
Elliptic Curves
The Tangent/Secant Process
Group Structure on an Elliptic Curve
Applications to Diophantine Equations

Elliptic Functions
Trigonometry Meets Diophantus
Elliptic Functions
Legendre and Weierstrass
Modular Functions

Wiles's Strategy and Recent Developments
The Frey Elliptic Curve
The Taniyama–Shimura–Weil Conjecture
Sketch Proof of Fermat's Last Theorem
Recent Developments

Quadratic Residues
Quadratic Equations in Zm
The Units of Zm
Quadratic Residues

Dirichlet’s Units Theorem
Logarithmic Space
Embedding the Unit Group in Logarithmic Space
Dirichlet's Theorem



Exercises appear at the end of each chapter.

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Ian Stewart is an emeritus professor of mathematics at the University of Warwick and a fellow of the Royal Society. Dr. Stewart has been a recipient of many honors, including the Royal Society’s Faraday Medal, the IMA Gold Medal, the AAAS Public Understanding of Science and Technology Award, and the LMS/IMA Zeeman Medal. He has published more than 180 scientific papers and numerous books, including several bestsellers co-authored with Terry Pratchett and Jack Cohen that combine fantasy with nonfiction.

David Tall is an emeritus professor of mathematical thinking at the University of Warwick. Dr. Tall has published numerous mathematics textbooks and more than 200 papers on mathematics and mathematics education. His research interests include cognitive theory, algebra, visualization, mathematical thinking, and mathematics education.


"It is the discussion of [Fermat’s Last Theorem], I think, that sets this book apart from others — there are a number of other texts that introduce algebraic number theory, but I don’t know of any others that combine that material with the kind of detailed exposition of FLT that is found here...To summarize and conclude: this is an interesting and attractive book. It would make an attractive text for an early graduate course on algebraic number theory, as well as a nice source of information for people interested in FLT, and especially its connections with algebraic numbers."
Dr. Mark Hunacek, MAA Reviews, June 2016

Praise for Previous Editions
"The book remains, as before, an extremely attractive introduction to algebraic number theory from the ideal-theoretic perspective."
—Andrew Bremner, Mathematical Reviews, February 2003