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• Cover Page
• Half-Title Page
• Title Page
• Copyright Page
• Contents
• Acknowledgements
• Preface to the Fifth Edition
• Historical Introduction
• 1 Classical Algebra
• 1.1 Complex Numbers
• 1.2 Subfields and Subrings of the Complex Numbers
• 1.3 Solving Equations
• 1.4 Solution by Radicals
• 2 The Fundamental Theorem of Algebra
• 2.1 Polynomials
• 2.2 Fundamental Theorem of Algebra
• 2.3 Implications
• 3 Factorisation of Polynomials
• 3.1 The Euclidean Algorithm
• 3.2 Irreducibility
• 3.3 Gauss’s Lemma
• 3.4 Eisenstein’s Criterion
• 3.5 Reduction Modulo p
• 3.6 Zeros of Polynomials
• 4 Field Extensions
• 4.1 Field Extensions
• 4.2 Rational Expressions
• 4.3 Simple Extensions
• 5 Simple Extensions
• 5.1 Algebraic and Transcendental Extensions
• 5.2 The Minimal Polynomial
• 5.3 Simple Algebraic Extensions
• 5.4 Classifying Simple Extensions
• 6 The Degree of an Extension
• 6.1 Definition of the Degree
• 6.2 The Tower Law
• 6.3 Primitive Element Theorem
• 7 Ruler-and-Compass Constructions
• 7.1 Approximate Constructions and More General Instruments
• 7.2 Constructions in ℂ
• 7.3 Specific Constructions
• 7.4 Impossibility Proofs
• 7.5 Construction from a Given Set of Points
• 8 The Idea behind Galois Theory
• 8.1 A First Look at Galois Theory
• 8.2 Galois Groups According to Galois
• 8.3 How to Use the Galois Group
• 8.4 The Abstract Setting
• 8.5 Polynomials and Extensions
• 8.6 The Galois Correspondence
• 8.7 Diet Galois
• 8.8 Natural Irrationalities
• 9 Normality and Separability
• 9.1 Splitting Fields
• 9.2 Normality
• 9.3 Separability
• 10 Counting Principles
• 10.1 Linear Independence of Monomorphisms
• 11 Field Automorphisms
• 11.1 K-Monomorphisms
• 11.2 Normal Closures
• 12 The Galois Correspondence
• 12.1 The Fundamental Theorem of Galois Theory
• 13 Worked Examples
• 13.1 Examples of Galois Groups
• 13.2 Discussion
• 14 Solubility and Simplicity
• 14.1 Soluble Groups
• 14.2 Simple Groups
• 14.3 Cauchy’s Theorem
• 15 Solution by Radicals
• 15.1 Radical Extensions
• 15.2 An Insoluble Quintic
• 15.3 Other Methods
• 16 Abstract Rings and Fields
• 16.1 Rings and Fields
• 16.2 General Properties of Rings and Fields
• 16.3 Polynomials Over General Rings
• 16.4 The Characteristic of a Field
• 16.5 Integral Domains
• 17 Abstract Field Extensions and Galois Groups
• 17.1 Minimal Polynomials
• 17.2 Simple Algebraic Extensions
• 17.3 Splitting Fields
• 17.4 Normality
• 17.5 Separability
• 17.6 Galois Theory for Abstract Fields
• 17.7 Conjugates and Minimal Polynomials
• 17.8 The Primitive Element Theorem
• 17.9 Algebraic Closure of a Field
• 18 The General Polynomial Equation
• 18.1 Transcendence Degree
• 18.2 Elementary Symmetric Polynomials
• 18.3 The General Polynomial
• 18.4 Cyclic Extensions
• 18.5 Solving Equations of Degree Four or Less
• 18.6 Explicit Formulas
• 19 Finite Fields
• 19.1 Structure of Finite Fields
• 19.2 The Multiplicative Group
• 19.3 Counterexample to the Primitive Element Theorem
• 19.4 Application to Solitaire
• 20 Regular Polygons
• 20.1 What Euclid Knew
• 20.2 Which Constructions are Possible?
• 20.3 Regular Polygons
• 20.4 Fermat Numbers
• 20.5 How to Construct a Regular 17-gon
• 21 Circle Division
• 21.1 Genuine Radicals
• 21.2 Fifth Roots Revisited
• 21.3 Vandermonde Revisited
• 21.4 The General Case
• 21.5 Cyclotomic Polynomials
• 21.6 Galois Group of ℚ(ζ)/ℚ
• 21.7 Constructions Using a Trisector
• 22 Calculating Galois Groups
• 22.1 Transitive Subgroups
• 22.2 Bare Hands on the Cubic
• 22.3 The Discriminant
• 22.4 General Algorithm for the Galois Group
• 23 Algebraically Closed Fields
• 23.1 Ordered Fields and Their Extensions
• 23.2 Sylow’s Theorem
• 23.3 The Algebraic Proof
• 24 Transcendental Numbers
• 24.1 Irrationality
• 24.2 Transcendence of e
• 24.3 Transcendence of π
• 25 What Did Galois Do or Know?
• 25.1 List of the Relevant Material
• 25.2 The First Memoir
• 25.3 What Galois Proved
• 25.4 What is Galois up to?
• 25.5 Alternating Groups, Especially A5
• 25.6 Simple Groups Known to Galois
• 25.7 Speculations about Proofs
• 25.8 A5 is Unique
• 26 Further Directions
• 26.1 Inverse Galois Problem
• 26.2 Differential Galois Theory
• 26.3 p-adic Numbers
• References
• Index