# Algebraic Number Theory and Fermat’s Last Theorem, 4th Edition

## 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

Part I. Algebraic Methods

Chapter 1. Algebraic Background

Chapter 2. Algebraic Numbers

Chapter 3. Quadratic And Cyclotomic Fields

Chapter 4. Factorization Into Irreducibles

Chapter 5. Ideals

Part II. Geometric Methods

Chapter 6. Lattices

Chapter 7. Minkowski’S Theorem

Chapter 8. Geometric Representation Of Algebraic Numbers

Chapter 9. Class-Group And Class-Number

Part III. Number-Theoretic Applications

Chapter 10. Computational Methods

Chapter 11. Kummer’S Special Case Of Fermat’S Last Theorem

Chapter 12. The Path To The Final Breakthrough

Chapter 13. Elliptic Curves

Chapter 14. Elliptic Functions

Chapter 15. Wiles’S Strategy And Recent Developments

Part IV. Appendices

Appendix A. Quadratic Residues

Appendix B. Dirichlet’S Units Theorem

## Book Details

- Title: Algebraic Number Theory and Fermat’s Last Theorem, 4th Edition
- Author: David Tall, Ian Stewart
- Length: 342 pages
- Edition: 4
- Language: English
- Publisher: Chapman and Hall/CRC
- Publication Date: 2015-10-13
- ISBN-10: 1498738397
- ISBN-13: 9781498738392

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