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
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.
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.
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( ) : 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