book

Galois Theory, 5th Edition

Name: Galois Theory, 5th Edition
Author: Ian Stewart
ISBN: 9781000644081

by Ian Stewart

September 2022

Intermediate to advanced

371 pages

8h 44m

English

Chapman and Hall/CRC

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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 Numbers1.2 Subfields and Subrings of the Complex Numbers1.3 Solving Equations1.4 Solution by Radicals
2 The Fundamental Theorem of Algebra
2.1 Polynomials2.2 Fundamental Theorem of Algebra2.3 Implications

3 Factorisation of Polynomials
3.1 The Euclidean Algorithm3.2 Irreducibility3.3 Gauss's Lemma3.4 Eisenstein's Criterion3.5 Reduction Modulo p3.6 Zeros of Polynomials
4 Field Extensions
4.1 Field Extensions4.2 Rational Expressions4.3 Simple Extensions
5 Simple Extensions
5.1 Algebraic and Transcendental Extensions5.2 The Minimal Polynomial5.3 Simple Algebraic Extensions5.4 Classifying Simple Extensions
6 The Degree of an Extension
6.1 Definition of the Degree6.2 The Tower Law6.3 Primitive Element Theorem
7 Ruler-and-Compass Constructions
7.1 Approximate Constructions and More General Instruments7.2 Constructions in ℂ7.3 Specific Constructions7.4 Impossibility Proofs7.5 Construction from a Given Set of Points
8 The Idea behind Galois Theory
8.1 A First Look at Galois Theory8.2 Galois Groups According to Galois8.3 How to Use the Galois Group8.4 The Abstract Setting8.5 Polynomials and Extensions8.6 The Galois Correspondence8.7 Diet Galois8.8 Natural Irrationalities
9 Normality and Separability
9.1 Splitting Fields9.2 Normality9.3 Separability
10 Counting Principles
10.1 Linear Independence of Monomorphisms
11 Field Automorphisms
11.1 K-Monomorphisms11.2 Normal Closures
12 The Galois Correspondence
12.1 The Fundamental Theorem of Galois Theory
13 Worked Examples
13.1 Examples of Galois Groups13.2 Discussion
14 Solubility and Simplicity
14.1 Soluble Groups14.2 Simple Groups14.3 Cauchy's Theorem
15 Solution by Radicals
15.1 Radical Extensions15.2 An Insoluble Quintic15.3 Other Methods
16 Abstract Rings and Fields
16.1 Rings and Fields16.2 General Properties of Rings and Fields16.3 Polynomials Over General Rings16.4 The Characteristic of a Field16.5 Integral Domains
17 Abstract Field Extensions and Galois Groups
17.1 Minimal Polynomials17.2 Simple Algebraic Extensions17.3 Splitting Fields17.4 Normality17.5 Separability17.6 Galois Theory for Abstract Fields17.7 Conjugates and Minimal Polynomials17.8 The Primitive Element Theorem17.9 Algebraic Closure of a Field
18 The General Polynomial Equation
18.1 Transcendence Degree18.2 Elementary Symmetric Polynomials18.3 The General Polynomial18.4 Cyclic Extensions18.5 Solving Equations of Degree Four or Less18.6 Explicit Formulas
19 Finite Fields
19.1 Structure of Finite Fields19.2 The Multiplicative Group19.3 Counterexample to the Primitive Element Theorem19.4 Application to Solitaire
20 Regular Polygons
20.1 What Euclid Knew20.2 Which Constructions are Possible?20.3 Regular Polygons20.4 Fermat Numbers20.5 How to Construct a Regular 17-gon
21 Circle Division
21.1 Genuine Radicals21.2 Fifth Roots Revisited21.3 Vandermonde Revisited21.4 The General Case21.5 Cyclotomic Polynomials21.6 Galois Group of ℚ(ζ)/ℚ21.7 Constructions Using a Trisector
22 Calculating Galois Groups
22.1 Transitive Subgroups22.2 Bare Hands on the Cubic22.3 The Discriminant22.4 General Algorithm for the Galois Group
23 Algebraically Closed Fields
23.1 Ordered Fields and Their Extensions23.2 Sylow's Theorem23.3 The Algebraic Proof
24 Transcendental Numbers
24.1 Irrationality24.2 Transcendence of e24.3 Transcendence of π
25 What Did Galois Do or Know?
25.1 List of the Relevant Material25.2 The First Memoir25.3 What Galois Proved25.4 What is Galois up to?25.5 Alternating Groups, Especially A525.6 Simple Groups Known to Galois25.7 Speculations about Proofs25.8 A5 is Unique
26 Further Directions
26.1 Inverse Galois Problem26.2 Differential Galois Theory26.3 p-adic Numbers
References
Index

Overview

Since 1973, Galois Theory has been educating undergraduate students on Galois groups and classical Galois theory. In Galois Theory, Fifth Edition, mathematician and popular science author Ian Stewart updates this well-established textbook for today’s algebra students.

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