Acknowledgements
Preface to the First Edition
Preface to the Second Edition
Preface to the Third Edition
Preface to the Fourth 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
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
l 1.2 Normal Closures
12 The Galois Correspondence
12.1 The Fundamental Theorem of Galois Theory
13 A Worked Example
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
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
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
19 Finite Fields
19.1 Structure of Finite Fields
19.2 The Multiplicative Group
19.3 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 Draw 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 ofQ(ζ) :Q
21.7 The Technical Lemma
21.8 More on Cyclotomic Polynomials
21.9 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, Espely A5
25.6 Simple Groups Known to Galois
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