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.

chapter |8 pages

The Origins of Algebraic Number Theory

part I|2 pages

Algebraic Methods

chapter 1|26 pages

Algebraic Background

chapter 2|26 pages

Algebraic Numbers

chapter 3|12 pages

Quadratic and Cyclotomic Fields

chapter 4|28 pages

Factorization into Irreducibles

chapter 5|26 pages


part II|2 pages

Geometric Methods

chapter 6|8 pages


chapter 7|6 pages

Minkowski’s Theorem

chapter 9|16 pages

Class-Group and Class-Number

part III|2 pages

Number-Theoretic Applications

chapter 10|14 pages

Computational Methods

chapter 12|14 pages

The Path to the Final Breakthrough

chapter 13|20 pages

Elliptic Curves

chapter 14|24 pages

Elliptic Functions

chapter 15|18 pages

Wiles’s Strategy and Recent Developments

chapter IV|32 pages