Galois Theory

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