Galois Theory, Fourth Edition: 4th Edition (Paperback) book cover

Galois Theory, Fourth Edition

By Ian Nicholas Stewart

© 2015 – Chapman and Hall/CRC

344 pages | 29 B/W Illus.

Comp Exam Copy
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Paperback: 9781482245820
pub: 2015-03-06
Hardback: 9781138401709
pub: 2017-07-27

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Since 1973, Galois Theory has been educating undergraduate students on Galois groups and classical Galois theory. In Galois Theory, Fourth Edition, mathematician and popular science author Ian Stewart updates this well-established textbook for today’s algebra students.

New to the Fourth Edition

  • The replacement of the topological proof of the fundamental theorem of algebra with a simple and plausible result from point-set topology and estimates that will be familiar to anyone who has taken a first course in analysis
  • Revised chapter on ruler-and-compass constructions that results in a more elegant theory and simpler proofs
  • A section on constructions using an angle-trisector since it is an intriguing and direct application of the methods developed
  • A new chapter that takes a retrospective look at what Galois actually did compared to what many assume he did
  • Updated references

This bestseller continues to deliver a rigorous yet engaging treatment of the subject while keeping pace with current educational requirements. More than 200 exercises and a wealth of historical notes augment the proofs, formulas, and theorems.


"… this book remains a highly recommended introduction to Galois theory along the more classical lines. It contains many exercises and a wealth of examples, including a pretty application of finite fields to the game solitaire. … provides readers with insight and historical perspective; it is written for readers who would like to understand this central part of basic algebra rather than for those whose only aim is collecting credit points."

Zentralblatt MATH 1322

Praise for the Third Edition:

"This edition preserves and even extends one of the most popular features of the original edition: the historical introduction and the story of the fatal duel of Evariste Galois. … These historical notes should be of interest to students as well as mathematicians in general. … after more than 30 years, Ian Stewart’s Galois Theory remains a valuable textbook for algebra undergraduate students."

Zentralblatt MATH, 1049

"The penultimate chapter is about algebraically closed fields and the last chapter, on transcendental numbers, contains ‘what-every-mathematician-should-see-at-least-once,’ the proof of transcendence of pi. … The book is designed for second- and third-year undergraduate courses. I will certainly use it."

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Table of Contents

Classical Algebra

Complex Numbers

Subfields and Subrings of the Complex Numbers

Solving Equations

Solution by Radicals

The Fundamental Theorem of Algebra


Fundamental Theorem of Algebra


Factorisation of Polynomials

The Euclidean Algorithm


Gauss’s Lemma

Eisenstein’s Criterion

Reduction Modulo p

Zeros of Polynomials

Field Extensions

Field Extensions

Rational Expressions

Simple Extensions

Simple Extensions

Algebraic and Transcendental Extensions

The Minimal Polynomial

Simple Algebraic Extensions

Classifying Simple Extensions

The Degree of an Extension

Definition of the Degree

The Tower Law

Ruler-and-Compass Constructions

Approximate Constructions and More General Instruments

Constructions in C

Specific Constructions

Impossibility Proofs

Construction from a Given Set of Points

The Idea behind Galois Theory

A First Look at Galois Theory

Galois Groups According to Galois

How to Use the Galois Group

The Abstract Setting

Polynomials and Extensions

The Galois Correspondence

Diet Galois

Natural Irrationalities

Normality and Separability

Splitting Fields



Counting Principles

Linear Independence of Monomorphisms

Field Automorphisms


Normal Closures

The Galois Correspondence

The Fundamental Theorem of Galois Theory

A Worked Example

Solubility and Simplicity

Soluble Groups

Simple Groups

Cauchy’s Theorem

Solution by Radicals

Radical Extensions

An Insoluble Quintic

Other Methods

Abstract Rings and Fields

Rings and Fields

General Properties of Rings and Fields

Polynomials over General Rings

The Characteristic of a Field

Integral Domains

Abstract Field Extensions

Minimal Polynomials

Simple Algebraic Extensions .

Splitting Fields



Galois Theory for Abstract Fields

The General Polynomial Equation

Transcendence Degree

Elementary Symmetric Polynomials

The General Polynomial

Cyclic Extensions

Solving Equations of Degree Four or Less

Finite Fields

Structure of Finite Fields

The Multiplicative Group

Application to Solitaire

Regular Polygons

What Euclid Knew

Which Constructions Are Possible?

Regular Polygons

Fermat Numbers

How to Draw a Regular 17-gon

Circle Division

Genuine Radicals

Fifth Roots Revisited

Vandermonde Revisited

The General Case

Cyclotomic Polynomials

Galois Group of Q(ζ) : Q

The Technical Lemma

More on Cyclotomic Polynomials

Constructions Using a Trisector

Calculating Galois Groups

Transitive Subgroups

Bare Hands on the Cubic

The Discriminant

General Algorithm for the Galois Group

Algebraically Closed Fields

Ordered Fields and Their Extensions

Sylow’s Theorem

The Algebraic Proof

Transcendental Numbers


Transcendence of e

Transcendence of pi

What Did Galois Do or Know?

List of the Relevant Material

The First Memoir

What Galois Proved

What Is Galois up to?

Alternating Groups, Especially A5

Simple Groups Known to Galois

Speculations about Proofs



About the Author

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.

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Algebra / General
MATHEMATICS / Number Theory