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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, Especially A5 25.6 Simple Groups Known to Galois
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