Galois Theory

Cover
Half Title
Title Page
Copyright Page
Dedication
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(ʕ)/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