StewartGaloisTheoryContents.pdf

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Galois Theory: Third Edition 
Ian Stewart 
 
Table of Contents 
 
Preface to the First Edition …………………………………………………………..… vii 
Preface to the Second Edition ………………………………………………………..…. ix 
Preface to the Third Edition …………………………………………………………..… xi 
Illustration Acknowledgments …………………………………………………..…...… xv 
Historical Introduction ……………………………………………………………...… xvii 
 
Chapter 1: Classical Algebra …………………………………………………..…...…. 1 
 1.1: Complex Numbers ……………………………………...…………..….....… 2 
 1.2: Subfields and Subrings of the Complex Numbers …...………………….….. 2 
 1.3: Solving Equations …………………...……………………..………...…...… 5 
 1.4: Solution by Radicals …………...…………………………………….…...… 6 
 1.4.1: Linear Equations …………...…………………….………...…...… 7 
 1.4.2: Quadratic Equations ………………………...…….………..…...… 7 
 1.4.3: Cubic Equations ……………………………………………..……..7 
 1.4.4: Peculiarities of Cardano’s Formula …………...………………..… 9 
 1.4.5: Quartic Equations …………………………..…………………… 10 
 1.4.6: Quintic Equations ……………………………..………………… 12 
 
Chapter 2: The Fundamental Theorem of Algebra ………………………………… 17 
 2.1: Polynomials …………………………………………...…………………... 17 
 2.2: Fundamental Theorem of Algebra ……………………...……………….… 21 
 2.3: Implications ……………………………………………..………………… 26 
 
Chapter 3: Factorization of Polynomials ……………………………………………. 31 
 3.1: The Euclidean Algorithm ……………………………......………………… 31 
 3.2: Irreducibility ……………………………………………...…………..…… 36 
 3.3: Gauss’ Lemma ……………………………………………...…….…..…… 39 
 3.4: Eisenstein’s Criterion ………………………………………...…...…..…… 40 
 3.5: Reduction Modulo p ……………………………………….......……..…… 42 
 3.6: Zeros of Polynomials ……………………………………...……...…..…… 44 
 
Chapter 4: Field Extensions ……………………………………………………..…… 49 
 4.1: Field Extensions …………………………………………...………….…… 49 
 4.2: Rational Expressions ……………………………………………...…..…… 53 
 4.3: Simple Extensions ………………………………………………...…..…… 54 
 
Chapter 5: Simple Extensions …………………………………….……………..…… 57 
 5.1: Algebraic and Transcendental Extensions …………….………...………… 57 
 5.2: The Minimal Polynomial ………………………...…..………...……..…… 58 
 5.3: Simple Algebraic Extensions ………………………………...…...…..…… 60 
 5.4: Classifying Simple Extensions ………………………………...…...…..…. 62 
Chapter 6: The Degree of an Extension …………..………………………….……… 67 
 6.1: Definition of the Degree …………………………………….………..…… 67 
 6.2: The Tower Law …………………………..........……….……………..…… 68 
 
Chapter 7: Ruler-and-Compass Constructions …………..………………………… 75 
 7.1: Algebraic Formulation ……………………………........……………..…… 76 
 7.2: Impossibility Proofs …………………………………....……………..…… 80 
 
Chapter 8: The Idea Behind Galois Theory ………………………...………….…… 85 
 8.1: A First Look at Galois Theory ……………………………...………...…… 85 
 8.2: Galois Groups According to Galois …………………………………..…… 86 
 8.3: … And How to Use Them ……………………………………..……..…… 88 
 8.4: The Abstract Setting ………………………………….………..……..…… 89 
 8.5: Polynomials and Extensions ……………………………………...…..…… 90 
 8.6: The Galois Correspondence …….……………………………….………… 92 
 8.7: Diet Galois ……………………….............…………….……………..…… 94 
 8.8: Natural Irrationalities …………………….…………….……………..…… 99 
 
Chapter 9: Normality and Separability ……………..………………………...…… 107 
 9.1: Splitting Fields …………………………….....……….……………..…… 107 
 9.2: Normality ……………………………………......………........……..…… 110 
 9.3: Separability …………………………………….……………............…… 112 
 
Chapter 10: Counting Principles ……….……………………….……………..…… 117 
 10.1: Linear Independence of Monomorphisms …………..……………..…… 117 
 
Chapter 11: Field Automorphisms …………………...………………………..…… 125 
 11.1: K-Monomorphisms ………………………………….……………..…… 125 
 11.2: Normal Closures …………………………………….……………..…… 127 
 
Chapter 12: The Galois Correspondence ….………………………………….…… 133 
 12.1: The Fundamental Theorem …………….…………………………..…… 133 
 
Chapter 13: A Worked Example ………………….…………….……………..…… 137 
 
Chapter 14: Solubility and Simplicity ……………….…………………….…..…… 143 
 14.1: Soluble Groups …………………………..………….……………..…… 143 
 14.2: Simple Groups …………………………...………….……………..…… 146 
 14.3: Cauchy’s Theorem ………………….……………….……………..…… 149 
 
Chapter 15: Solution by Radicals …………………………………….………..…… 153 
 15.1: Radical Extensions …………………………………..……………..…… 153 
 15.2: An Insoluble Quintic ………………………………...……………..…… 158 
 15.3: Other Methods …………………………………….…...…………..…… 160 
 
 
Chapter 16: Abstract Rings and Fields ………………………………………..…… 163 
 16.1: Rings and Fields …………………………………….……………..…… 163 
 16.2: General Properties of Rings and Fields ………………………………… 166 
 16.3: Polynomials Over General Rings ………………….…..…………..…… 168 
 16.4: The Characteristic of a Field ……………………….…..…………..…… 169 
 16.5: Integral Domains …………………………………….……………..…… 171 
 
Chapter 17: Abstract Field Extensions ………………………………………..…… 177 
 17.1: Minimal Polynomials ……………………………….….…………..…… 177 
 17.2: Simple Algebraic Extensions ……………………………...……….…… 178 
 17.3: Splitting Fields …………………………………….………...……..…… 180 
 17.5: Separability …………………………………….……………..........…… 182 
 17.6: Galois Theory for Abstract Fields ………………………………....…… 187 
 
Chapter 18: The General Polynomial ……………………….………………...…… 191 
 18.1: Transcendence Degrees ………………………….……………..….....… 191 
 18.2: Elementary Symmetric Polynomials ……………………………..…..… 193 
 18.3: The General Polynomial …………………………………….……..…… 194 
 18.4: Cyclic Extensions ……………………………….……………........…… 197 
 18.5: Solving Quartic Extensions …………....…………………………..…… 201 
 18.5.4: Quartic Equations ……………...........…….……………..…… 202 
 18.5.5: Explicit Formulas …………………………………….…..…… 203 
 
Chapter 19: Regular Polygons …………………….…………….……………..…… 209 
 19.1: What Euclid Knew ……………….………………….……………..…… 209 
 19.2: Which Constructions are Possible? .………………………………..…… 212 
 19.3: Regular Polygons ……………………...…………….……………..…… 215 
 19.4: Fermat Numbers …………………………………….……………..…… 219 
 19.5: How to Draw a Regular 17-Gon …….……………………………..…… 220 
 
Chapter 20: Finite Fields ………………………......…………….……………..…… 227 
 20.1: Structure of Finite Fields …………………………………………..…… 227 
 20.2: The Multiplicative Group ………...………………………………..…… 228 
 20.3: Application to Solitaire ……………..……………………….……..…… 230 
 
Chapter 21: Circle Division …………………………………….………..……..…… 233 
 21.1: Nontrivial Radicals ………………………………….……………..…… 234 
 21.2: Fifth Roots Revisited ……………………………….…….………..…… 236 
 21.3: Vandermonde Revisited ………………………………….….……..…… 239 
 21.4: The General Case ………………………………….……...………..…… 240 
 21.5: Cyclotomic Polynomials …………………………………….……..…… 244 
 21.6: The Technical Lemma ………………………………...…….……..…… 247 
 21.7: More on Cyclotomic Polynomials …………………..……………..…… 248 
 
 
 
Chapter 22: Calculating Galois Groups …………………………..…………..…… 251 
 22.1: Transitive Subgroups ………………………………..……………..…… 251 
 22.2: Bare Hands on the Cubic ………………………….…...…………..…… 252 
 22.3: The Discriminant ………………………………….…...…………..…… 255 
 22.4: General Algorithm ………………………………….….…………..…… 256 
 
Chapter 23: Algebraically Closed Fields …………………………….………..…… 261 
 23.1: Ordered Fields and their Extensions ………………………...……..…… 261 
 23.2: Sylow’s Theorem ………………………………….……...………..…… 263 
 23.3: The Algebraic Proof ……………………………….………...……..…… 265 
 
Chapter 24: Transcendental Numbers …………………………………….…..…… 269 
 24.1: Irrationality ………………….………………….………….......…..…… 270 
 24.2: Transcendence of e …………………………….……………......……… 272 
 24.3: Transcendence of π …………………………………….…………..…… 274 
 
References ………….......………………....………….……........………..............…… 279 
Index ……………..........…………….....……........…….…….....…….......…..…....… 283