Galois Theory
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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 wellestablished 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 pointset topology and estimates that will be familiar to anyone who has taken a first course in analysis
 Revised chapter on rulerandcompass constructions that results in a more elegant theory and simpler proofs
 A section on constructions using an angletrisector 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
RulerandCompass 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
KMonomorphisms
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 17gon
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 A_{5}
Simple Groups Known to Galois
Speculations about Proofs
References
Index
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 coauthored 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 1322Praise 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 ‘whateverymathematicianshouldseeatleastonce,’ the proof of transcendence of pi. … The book is designed for second and thirdyear undergraduate courses. I will certainly use it."
—EMS Newsletter
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