Galois Theory, 5th Edition

Table of contents

1. Cover Page
2. Half-Title Page
3. Title Page
4. Copyright Page
5. Contents
6. Acknowledgements
7. Preface to the Fifth Edition
8. Historical Introduction
9. 1 Classical Algebra
   1. 1.1 Complex Numbers
   2. 1.2 Subfields and Subrings of the Complex Numbers
   3. 1.3 Solving Equations
   4. 1.4 Solution by Radicals
10. 2 The Fundamental Theorem of Algebra
    1. 2.1 Polynomials
    2. 2.2 Fundamental Theorem of Algebra
    3. 2.3 Implications
11. 3 Factorisation of Polynomials
    1. 3.1 The Euclidean Algorithm
    2. 3.2 Irreducibility
    3. 3.3 Gauss's Lemma
    4. 3.4 Eisenstein's Criterion
    5. 3.5 Reduction Modulo p
    6. 3.6 Zeros of Polynomials
12. 4 Field Extensions
    1. 4.1 Field Extensions
    2. 4.2 Rational Expressions
    3. 4.3 Simple Extensions
13. 5 Simple Extensions
    1. 5.1 Algebraic and Transcendental Extensions
    2. 5.2 The Minimal Polynomial
    3. 5.3 Simple Algebraic Extensions
    4. 5.4 Classifying Simple Extensions
14. 6 The Degree of an Extension
    1. 6.1 Definition of the Degree
    2. 6.2 The Tower Law
    3. 6.3 Primitive Element Theorem
15. 7 Ruler-and-Compass Constructions
    1. 7.1 Approximate Constructions and More General Instruments
    2. 7.2 Constructions in ℂ
    3. 7.3 Specific Constructions
    4. 7.4 Impossibility Proofs
    5. 7.5 Construction from a Given Set of Points
16. 8 The Idea behind Galois Theory
    1. 8.1 A First Look at Galois Theory
    2. 8.2 Galois Groups According to Galois
    3. 8.3 How to Use the Galois Group
    4. 8.4 The Abstract Setting
    5. 8.5 Polynomials and Extensions
    6. 8.6 The Galois Correspondence
    7. 8.7 Diet Galois
    8. 8.8 Natural Irrationalities
17. 9 Normality and Separability
    1. 9.1 Splitting Fields
    2. 9.2 Normality
    3. 9.3 Separability
18. 10 Counting Principles
    1. 10.1 Linear Independence of Monomorphisms
19. 11 Field Automorphisms
    1. 11.1 K-Monomorphisms
    2. 11.2 Normal Closures
20. 12 The Galois Correspondence
    1. 12.1 The Fundamental Theorem of Galois Theory
21. 13 Worked Examples
    1. 13.1 Examples of Galois Groups
    2. 13.2 Discussion
22. 14 Solubility and Simplicity
    1. 14.1 Soluble Groups
    2. 14.2 Simple Groups
    3. 14.3 Cauchy's Theorem
23. 15 Solution by Radicals
    1. 15.1 Radical Extensions
    2. 15.2 An Insoluble Quintic
    3. 15.3 Other Methods
24. 16 Abstract Rings and Fields
    1. 16.1 Rings and Fields
    2. 16.2 General Properties of Rings and Fields
    3. 16.3 Polynomials Over General Rings
    4. 16.4 The Characteristic of a Field
    5. 16.5 Integral Domains
25. 17 Abstract Field Extensions and Galois Groups
    1. 17.1 Minimal Polynomials
    2. 17.2 Simple Algebraic Extensions
    3. 17.3 Splitting Fields
    4. 17.4 Normality
    5. 17.5 Separability
    6. 17.6 Galois Theory for Abstract Fields
    7. 17.7 Conjugates and Minimal Polynomials
    8. 17.8 The Primitive Element Theorem
    9. 17.9 Algebraic Closure of a Field
26. 18 The General Polynomial Equation
    1. 18.1 Transcendence Degree
    2. 18.2 Elementary Symmetric Polynomials
    3. 18.3 The General Polynomial
    4. 18.4 Cyclic Extensions
    5. 18.5 Solving Equations of Degree Four or Less
    6. 18.6 Explicit Formulas
27. 19 Finite Fields
    1. 19.1 Structure of Finite Fields
    2. 19.2 The Multiplicative Group
    3. 19.3 Counterexample to the Primitive Element Theorem
    4. 19.4 Application to Solitaire
28. 20 Regular Polygons
    1. 20.1 What Euclid Knew
    2. 20.2 Which Constructions are Possible?
    3. 20.3 Regular Polygons
    4. 20.4 Fermat Numbers
    5. 20.5 How to Construct a Regular 17-gon
29. 21 Circle Division
    1. 21.1 Genuine Radicals
    2. 21.2 Fifth Roots Revisited
    3. 21.3 Vandermonde Revisited
    4. 21.4 The General Case
    5. 21.5 Cyclotomic Polynomials
    6. 21.6 Galois Group of ℚ(ζ)/ℚ
    7. 21.7 Constructions Using a Trisector
30. 22 Calculating Galois Groups
    1. 22.1 Transitive Subgroups
    2. 22.2 Bare Hands on the Cubic
    3. 22.3 The Discriminant
    4. 22.4 General Algorithm for the Galois Group
31. 23 Algebraically Closed Fields
    1. 23.1 Ordered Fields and Their Extensions
    2. 23.2 Sylow's Theorem
    3. 23.3 The Algebraic Proof
32. 24 Transcendental Numbers
    1. 24.1 Irrationality
    2. 24.2 Transcendence of e
    3. 24.3 Transcendence of π
33. 25 What Did Galois Do or Know?
    1. 25.1 List of the Relevant Material
    2. 25.2 The First Memoir
    3. 25.3 What Galois Proved
    4. 25.4 What is Galois up to?
    5. 25.5 Alternating Groups, Especially A5
    6. 25.6 Simple Groups Known to Galois
    7. 25.7 Speculations about Proofs
    8. 25.8 A5 is Unique
34. 26 Further Directions
    1. 26.1 Inverse Galois Problem
    2. 26.2 Differential Galois Theory
    3. 26.3 p-adic Numbers
35. References
36. Index