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4th Edition

Galois Theory





ISBN 9781482245820
Published March 6, 2015 by Chapman and Hall/CRC
344 Pages - 29 B/W Illustrations

 
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Book Description

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.

Table of Contents

Classical Algebra
Complex Numbers
Subfields and Subrings of the Complex Numbers
Solving Equations
Solution by Radicals

The Fundamental Theorem of Algebra
Polynomials
Fundamental Theorem of Algebra
Implications

Factorisation of Polynomials
The Euclidean Algorithm
Irreducibility
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
Normality
Separability

Counting Principles
Linear Independence of Monomorphisms

Field Automorphisms
K-Monomorphisms
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
Normality
Separability
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
Irrationality
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

References
Index

...
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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.

Reviews

"… 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."
EMS Newsletter

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