# Algebraic Number Theory and Fermat's Last Theorem

## Preview

## 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

Modules

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

**Ideals **Historical Background

Prime Factorization of Ideals

The Norm of an Ideal

Nonunique Factorization in Cyclotomic Fields

** Geometric Methods Lattices **Lattices

The Quotient Torus

**Minkowski's Theorem**

Minkowski's Theorem

The Two-Squares Theorem

The Four-Squares Theorem

**Geometric Representation of Algebraic Numbers **The Space L

^{st}

**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

** AppendicesQuadratic Residues **Quadratic Equations in Z

_{m}The Units of Z

_{m}Quadratic Residues

**Dirichlet’s Units Theorem**

Introduction

Logarithmic Space

Embedding the Unit Group in Logarithmic Space

Dirichlet's Theorem

**Bibliography **

**Index**

*Exercises appear at the end of each chapter.*

## Author(s)

### Biography

**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.

## Reviews

"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