Graduate Texts in Mathematics 204 Editorial Board

S. Axler F.W. Gehring K.A. Ribet

Springer Science+Business Media, LLC

Graduate Texts in Mathematics

TAKEUTIlZARING.lntroduction to 34 SPITZER. Principles of Random Walk. Axiomatic Set Theory. 2nd ed. 2nded.

2 OxrOBY. Measure and Category. 2nd ed. 35 Al.ExANDERIWERMER. Several Complex 3 SCHAEFER. Topological Vector Spaces. Variables and Banach Algebras. 3rd ed.

2nded. 36 l<ELLEyINAMIOKA et al. Linear Topological 4 HILTON/STAMMBACH. A Course in Spaces.

Homological Algebra. 2nd ed. 37 MONK. Mathematical Logic. 5 MAc LANE. Categories for the Working 38 GRAUERT/FRlTZSCHE. Several Complex

Mathematician. 2nd ed. Variables. 6 HUGHEs/Pn>ER. Projective Planes. 39 ARVESON. An Invitation to C*-Algebras. 7 SERRE. A Course in Arithmetic. 40 KEMENY/SNELLIKNAPP. Denumerable 8 T AKEUTIlZARING. Axiomatic Set Theory. Markov Chains. 2nd ed. 9 HUMPHREYS. Introduction to Lie Algebras 41 APOSTOL. Modular Functions and

and Representation Theory. Dirichlet Series in Number Theory. 10 COHEN. A Course in Simple Homotopy 2nded.

Theory. 42 SERRE. Linear Representations of Finite II CONWAY. Functions of One Complex Groups.

Variable I. 2nd ed. 43 GILLMAN/JERISON. Rings of Continuous 12 BEALS. Advanced Mathematical Analysis. Functions. 13 ANDERSoN/FULLER. Rings and Categories 44 KENDIG. Elementary Algebraic Geometry.

of Modules. 2nd ed. 45 LoEVE. Probability Theory I. 4th ed. 14 GoLUBITSKy/GUILLEMIN. Stable Mappings 46 LoEVE. Probability Theory II. 4th ed.

and Their Singularities. 47 MorSE. Geometric Topology in 15 BERBERIAN. Lectures in Functional Dimensions 2 and 3.

Analysis and Operator Theory. 48 SACHs/WU. General Relativity for 16 WINTER. The Structure of Fields. Mathematicians. 17 ROSENBLATT. Random Processes. 2nd ed. 49 GRUENBERGlWEIR. Linear Geometry. 18 HALMos. Measure Theory. 2nded. 19 HALMos. A Hilbert Space Problem Book. 50 EDWARDS. Fennat's Last Theorem.

2nded. 51 KLINGENBERG. A Course in Differential 20 HUSEMOLLER. Fibre Bundles. 3rd ed. Geometry. 21 HUMPHREYS. Linear Algebraic Groups. 52 HARTSHORNE. Algebraic Geometry. 22 BARNEs/MAcK. An Algebraic Introduction 53 MANIN. A Course in Mathematical Logic.

to Mathematical Logic. 54 GRAVERlWATKlNS. Combinatorics with 23 GREUB. Linear Algebra. 4th ed. Emphasis on the Theory of Graphs. 24 HOLMES. Geometric Functional Analysis 55 BRoWN!PEARCY. Introduction to Operator

and Its Applications. Theory I: Elements of Functional Analysis. 25 HEWITT/STROMBERG. Real and Abstract 56 MAsSEY. Algebraic Topology: An

Analysis. Introduction. 26 MANEs. Algebraic Theories. 57 CRoWELLlFox. Introduction to Knot 27 l<ELLEy. General Topology. Theory. 28 ZARISKIISAMUEL. Commutative Algebra. 58 KOBLITZ. p-adic Numbers, p-adic

Vol.I. Analysis, and Zeta-Functions. 2nd ed. 29 ZARIsKIlSAMUEL. Commutative Algebra. 59 LANG. Cyclotomic Fields.

Vol.Il. 60 ARNOLD. Mathematical Methods in 30 JACOBSON. Lectures in Abstract Algebra I. Classical Mechanics. 2nd ed.

Basic Concepts. 61 WHITEHEAD. Elements of Homotopy 31 JACOBSON. Lectures in Abstract Algebra II. Theory.

Linear Algebra 62 KARGAPOLOv~AKov.Fundamenta1s

32 JACOBSON. Lectures in Abstract Algebra of the Theory of Groups. ill. Theory of Fields and Galois Theory. 63 BOLLOBAS. Graph Theory.

33 HIRsCH. Differential Topology. 64 EDWARDS. Fourier Series. Vol. I. 2nd ed.

(continued after index)

Jean-Pierre Escofier

Galois Theory

Translated by Leila Schneps

With 48 Illustrations

Springer

Jean-Pierre Escofier Translator Leila Schneps Institute Mathematiques de Rennes

Campus de Beaulieu Universite de Rennes 1

36 rue de ]' Orillon 75011 Paris

35042 Rennes Cedex France jean-pierre.escofier@univ-rennesl.fr

France leila.schneps@ens.fr

Editorial Board

S. Axler Mathematics Department San Francisco State

University San Francisco, CA 94132 USA

F.W. Gehring Mathematics Department East Hali University of Michigan Ann Arbor, MI 48109 USA

K.A. Ribet Mathematics Department University of California

at Berkeley Berkeley, CA 94720-3840 USA

Mathematics Subject Classification (2000): 11 R32, Il S20, 12F10, 13B05

Library of Congress Cataloging-in-Publieation Data Eseofier, Jean-Pierre.

Galois theory / Jean-Pierre Escofier. p. em. - (Graduate texts in mathematics; 204)

Includes bibliographical references and index. ISBN 978-1-4612-6558-0 ISBN 978-1-4613-0191-2 (eBook) DOI 10.1007/978-1-4613-0191-2 1. Galois theory. 1. Title. II. Series.

QA174.2 .E73 2000 5 1 2'.3-de2 1 00-041906

Printed on acid-free paper.

Translated from the Freneh Theorie de Galois, by Jean-Pierre Eseofier, first edition published by Masson, Paris, © 1997, and second edition published by Dunod, Paris, © 2000, 5, rue Laromiguiere, 75005 Paris, France.

© 2001 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 2001 Softcover reprint of the hardcover 1 st edition 2001 AII rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews ar scholarly analysis. Use in connection with any form of infor­mation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the forrner are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by Francine McNeill; manufacturing supervised by Joe Quatela. Photocomposed copy prepared from the translator' s TeX files.

9 8 7 654 3 2 1

ISBN 978-1-4612-6558-0 SPIN 1071 1904

Preface

This book begins with a sketch, in Chapters 1 and 2, of the study of alge­braic equations in ancient times (before the year 1600). After introducing symmetric polynomials in Chapter 3, we consider algebraic extensions of fi­nite degree contained in the field <C of complex numbers (to remain within a familiar framework) and develop the Galois theory for these fields in Chapters 4 to 8. The fundamental theorem of Galois theory, that is, the Galois correspondence between groups and field extensions, is contained in Chapter 8. In order to give a rounded aspect to this basic introduction of Galois theory, we also provide

• a digression on constructions with ruler and compass (Chapter 5),

• beautiful applications (Chapters 9 and 10), and

• a criterion for solvability of equations by radicals (Chapters 11 and 12).

Many of the results presented here generalize easily to arbitrary fields (at least in characteristic 0), or they can be adapted to extensions of infinite degree.

I could not write a book on Galois theory without some mention of the exceptional life of Evariste Galois (Chapter 13). The bibliography provides details on where to obtain further information about his life, as well as information on the moving story of Niels Abel.

After these chapters, we introduce finite fields (Chapter 14) and separable extensions (Chapter 15). Chapter 16 presents two topics of current research:

vi Preface

firstly, the inverse Galois problem, which asks whether all finite groups occur as Galois groups of finite extensions of Q and which we treat explicitly in one very simple case, and secondly, a method for computing Galois groups that can be programmed on a computer.

Most of the chapters contain exercises and problems. Some of the state­ments are for practice, or are taken from past examinations; others suggest interesting results beyond the scope of the text. Some solutions are given completely, others are sketchy, and certain solutions that would involve mathematics beyond the scope of the text are omitted completely.

Finally, this book contains a brief sketch of the history of Galois theory. I would like to thank the municipal library in Rennes for having allowed me to reproduce some fragments of its numerous treasures.

The entire book was written with its student readers in mind, and with constant, careful consideration of the question of what these students will remember of it several years from now.

lowe tremendous thanks to Annette Houdebine-Paugam, who helped me many times, and to Bernard Le Stum and Masson, who read the later versions of the text and suggested many corrections and alterations.

Jean-Pierre Escofier

May 1997

Contents

Preface v

1 Historical Aspects of the Resolution of Algebraic Equations 1 1.1 Approximating the Roots of an Equation ..... 1 1.2 Construction of Solutions by Intersections of Curves 2 1.3 Relations with Trigonometry ....... 2 1.4 Problems of Notation and Terminology. . 3 1.5 The Problem of Localization of the Roots 4 1.6 The Problem of the Existence of Roots. . 5 1. 7 The Problem of Algebraic Solutions of Equations 6

Toward Chapter 2 ... . . . . . . . . . . . . . . 7

2 Resolution of Quadratic, Cubic, and Quartic Equations 9 2.1 Second-Degree Equations 9

2.1.1 The Babylonians 9 2.1.2 The Greeks . . . . 11 2.1.3 The Arabs. . . . . 11 2.1.4 Use of Negative Numbers

2.2 Cubic Equations ........ . 2.2.1 The Greeks ....... . 2.2.2 Omar Khayyam and Sharaf ad Din at Tusi 2.2.3 Scipio del Ferro, Tartaglia, Cardan . . . . . 2.2.4 Algebraic Solution of the Cubic Equation . 2.2.5 First Computations with Complex Numbers. 2.2.6 Raffaele Bombelli ............... .

12 13 13 13 14 15 16 17

viii Contents

2.2.7 Fran<;ois Viete . 2.3 Quartic Equations .. .

Exercises for Chapter 2 Solutions to Some of the Exercises

18 18 19 22

3 Symmetric Polynomials 25 3.1 Symmetric Polynomials 25

3.1.1 Background... 25 3.1.2 Definitions ... 26

3.2 Elementary Symmetric Polynomials 27 3.2.1 Definition........... 27 3.2.2 The Product of the X - Xi; Relations Between Co-

efficients and Roots .................. 27 3.3 Symmetric Polynomials and Elementary Symmetric Polyno-

mials . . . . . . . . 29 3.3.1 Theorem . 29 3.3.2 Proposition 31 3.3.3 Proposition 32

3.4 Newton's Formulas 32 3.5 Resultant of Two Polynomials . 35

3.5.1 Definition....... 35 3.5.2 Proposition...... 35

3.6 Discriminant of a Polynomial 37 3.6.1 Definition. 37 3.6.2 Proposition...... 37 3.6.3 Formulas ....... 38 3.6.4 Polynomials with Real Coefficients: Real Roots and

Sign of the Discriminant . . 38 Exercises for Chapter 3 ...... 39 Solutions to Some of the Exercises 44

4 Field Extensions 4.1 Field Extensions

4.1.1 Definition 4.1.2 Proposition 4.1.3 The Degree of an Extension 4.1.4 Towers of Fields

4.2 The Tower Rule 4.2.1 Proposition ..

4.3 Generated Extensions 4.3.1 Proposition 4.3.2 Definition.. 4.3.3 Proposition.

4.4 Algebraic Elements . 4.4.1 Definition ..

51 51 51 52 52 52 53 53 54 54 55 55 55 55

Contents ix

4.4.2 Transcendental Numbers. . . . . . . . . . . . 55 4.4.3 Minimal Polynomial of an Algebraic Element 56 4.4.4 Definition.................... 56 4.4.5 Properties of the Minimal Polynomial . . . . 57 4.4.6 Proving the Irreducibility of a Polynomial in Z[X] 57

4.5 Algebraic Extensions . . . . . . . . . . . . . . . . . . . 59 4.5.1 Extensions Generated by an Algebraic Element 59 4.5.2 Properties of K[a] ..... 59 4.5.3 Definition.................. 60 4.5.4 Extensions of Finite Degree . . . . . . . . 60 4.5.5 Corollary: Towers of Algebraic Extensions 61

4.6 Algebraic Extensions Generated by n Elements 61 4.6.1 Notation.. 61 4.6.2 Proposition................ 61 4.6.3 Corollary . . . . . . . . . . . . . . . . . 62

4.7 Construction of an Extension by Adjoining a Root 62 4.7.1 Definition. 62 4.7.2 Proposition............ 4.7.3 Corollary . . . . . . . . . . . . . 4.7.4 Universal Property of K[XJI(P) Toward Chapters 5 and 6 . . . . . Exercises for Chapter 4 ..... . Solutions to Some of the Exercises

62 63 63 64 64 69

5 Constructions with Straightedge and Compass 79 5.1 Constructible Points . . . . . . . . . . . 79 5.2 Examples of Classical Constructions . . . . . . . 80

5.2.1 Projection of a Point onto a Line 80 5.2.2 Construction of an Orthonormal Basis from Two Points 80 5.2.3 Construction of a Line Parallel to a Given Line Pass-

ing Through a Point . . . . . . . . . . . . 81 5.3 Lemma ....................... 82 5.4 Coordinates of Points Constructible in One Step 82 5.5 A Necessary Condition for Constructibility ... 83 5.6 Two Problems More Than Two Thousand Years Old 84

5.6.1 Duplication of the Cube . . . . . . . 85 5.6.2 Trisection of the Angle. . . . . . . . 85

5.7 A Sufficient Condition for Constructibility . 85 Exercises for Chapter 5 ...... 87 Solutions to Some of the Exercises . . . . . 90

6 K-Homomorphisms 6.1 Conjugate Numbers 6.2 K-Homomorphisms.

6.2.1 Definitions .

93 93 94 94

x Contents

6.2.2 Properties.... ... . . . . . . 6.3 Algebraic Elements and K-Homomorphisms .

6.3.1 Proposition ....... . 6.3.2 Example..... .. .

6.4 Extensions of Embeddings into C 6.4.1 Definition 6.4.2 Proposition ...... . 6.4.3 Proposition ...... .

6.5 The Primitive Element Theorem 6.5.1 Theorem and Definition 6.5.2 Example..... ...

6.6 Linear Independence of K-Homomorphisms 6.6.1 Characters .. . .. " 6.6.2 Emil Artin's Theorem . . . . . 6.6.3 Corollary: Dedekind's Theorem Exercises for Chapter 6 .... Solutions to Some of the Exercises

94 95 95 96 97 97 97 98 99 99

100 101 101 101 102 102 103

7 Normal Extensions 107 7.1 Splitting Fields . . .. .......... 107

7.1.1 Definition................ 107 7.1.2 Splitting Field of a Cubic Polynomial 108

7.2 Normal Extensions . . ., ......... 108 7.3 Normal Extensions and K-Homomorphisms 109 7.4 Splitting Fields and Normal Extensions 109

7.4.1 Proposition ...... ...... 109 7.4.2 Converse .... ... ...... 110

7.5 Normal Extensions and Intermediate Extensions 110 7.6 Normal Closure. . 111

7.6.1 Definition 111 7.6.2 Proposition 111 7.6.3 Proposition 111

7.7 Splitting Fields: General Case . 112 Toward Chapter 8 ...... 113 Exercises for Chapter 7 ... 113 Solutions to Some of the Exercises 115

8 Galois Groups 119 8.1 Galois Groups . . . . . . . . . . . . . . . .. 119

8.1.1 The Galois Group of an Extension ........ 119 8.1.2 The Order of the Galois Group of a Normal Exten-

sion of Finite Degree. . . . . . . . . . . . . . . .. 120 8.1.3 The Galois Group of a Polynomial ......... 120 8.1.4 The Galois Group as a Subgroup of a Permutation

Group. . . . . . . . . . . . . . . . . . . . . " 120

Contents xi

8.1.5 A Short History of Groups 121 8.2 Fields of Invariants . . . . . . . . . 122

8.2.1 Definition and Proposition. 122 8.2.2 Emil Artin's Theorem . . . 122

8.3 The Example of Q [ij2,j]: First Part 124 8.4 Galois Groups and Intermediate Extensions 126 8.5 The Galois Correspondence . . . . . . . 126 8.6 The Example of Q [ ij2, j]: Second Part 128 8.7 The Example X4 + 2 . . . . . . 128

8.7.1 Dihedral Groups . . . . . . . 128 8.7.2 The Special Case of D4 ... 129 8.7.3 The Galois Group of X4 + 2 130 8.7.4 The Galois Correspondence . 130 8.7.5 Search for Minimal Polynomials. 132 Toward Chapters 9, 10, and 12 . . 133 Exercises for Chapter 8 ...... 133 Solutions to Some of the Exercises 139

9 Roots of Unity 149 9.1 The Group U(n) of Units of the Ring 7l./n7l. 149

9.1.1 Definition and Background 149 9.1.2 The Structure of U(n) . . 150

9.2 The Mobius Function ...... 151 9.2.1 Multiplicative Functions. 151 9.2.2 The Mobius Function .. 151 9.2.3 Proposition........ 151 9.2.4 The Mobius Inversion Formula 152

9.3 Roots of Unity . . . . . . . 153 9.3.1 n-th Roots of Unity 153 9.3.2 Proposition..... 153 9.3.3 Primitive Roots. . . 153 9.3.4 Properties of Primitive Roots 153

9.4 Cyclotomic Polynomials . . . . . . . 153 9.4.1 Definition........... 153 9.4.2 Properties of the Cyclotomic Polynomial 153

9.5 The Galois Group over Q of an Extension of Q by a Root of Unity . . . . . . . . . . . . . . . 156 Exercises for Chapter 9 ...... 157 Solutions to Some of the Exercises 163

10 Cyclic Extensions 179 10.1 Cyclic and Abelian Extensions ........ 179 10.2 Extensions by a Root and Cyclic Extensions. 179 10.3 Irreducibility of XP - a 180 10.4 Hilbert's Theorem 90. . . . . . . . . . . . . . 181

xii Contents

10.4.1 The Norm. . . . . . . . . . . . . . . . . . . . . . 181 10.4.2 Hilbert's Theorem 90 ............... 182

10.5 Extensions by a Root and Cyclic Extensions: Converse. 182 10.6 Lagrange Resolvents 183

10.6.1 Definition . . . . . . . . . . 183 10.6.2 Properties ......... 183

10.7 Resolution of the Cubic Equation 184 10.8 Solution of the Quartic Equation 186 10.9 Historical Commentary ...... 188

Exercises for Chapter 10 . . . . . . 188 Solutions to Some of the Exercises 190

11 Solvable Groups 11.1 First Definition

195 195

11.2 Derived or Commutator Subgroup 196 11.3 Second Definition of Solvability 196 11.4 Examples of Solvable Groups . . . 197 11.5 Third Definition .......... 197 11.6 The Group An Is Simple for n ;::: 5 198

11.6.1 Theorem .......... 198 11.6.2 An Is Not Solvable for n ;::: 5, Direct Proof. 199

11.7 Recent Results . . . . . . . . . . . 199 Exercises for Chapter 11 . . . . . . 200 Solutions to Some of the Exercises 203

12 Solvability of Equations by Radicals 207 12.1 Radical Extensions and Polynomials Solvable by Radicals 207

12.1.1 Radical Extensions ........ 207 12.1.2 Polynomials Solvable by Radicals. 208 12.1.3 First Construction . . . . . . . . . 208 12.1.4 Second Construction . . . . . . . . 208

12.2 If a Polynomial Is Solvable by Radicals, Its Galois Group Is Solvable . . . . . . . . . . . . . . . . . . . . . . . . 209

12.3 Example of a Polynomial Not Solvable by Radicals 209 12.4 The Converse of the Fundamental Criterion 210 12.5 The General Equation of Degree n . . . . . . . . . 210

12.5.1 Algebraically Independent Elements . . . . 210 12.5.2 Existence of Algebraically Independent Elements 211 12.5.3 The General Equation of Degree n . . . . . . . . 211 12.5.4 Galois Group of the General Equation of Degree n 211 Exercises for Chapter 12 . . . . . . 212 Solutions to Some of the Exercises . . . . . . . . . . . .. 214

13 The Life of Evariste Galois 219

Contents xiii

14 Finite Fields 227 14.1 Algebraically Closed Fields 227

14.1.1 Definition . . . . . . 227 14.1.2 Algebraic Closures . 228 14.1.3 Theorem (Steinitz, 1910) 228

14.2 Examples of Finite Fields . . 229 14.3 The Characteristic of a Field 229

14.3.1 Definition . . . . . 229 14.3.2 Properties . . . . . 229

14.4 Properties of Finite Fields 230 14.4.1 Proposition . . . . 230 14.4.2 The Frobenius Homomorphism 23i

14.5 Existence and Uniqueness of a Finite Field with pr Elements 231 14.5.1 Proposition . . . . . 231 14.5.2 Corollary . . . . . . . . . . . . . . . . . . 232

14.6 Extensions of Finite Fields. . . . . . . . . . . . . 233 14.7 Normality of a Finite Extension of Finite Fields. 233 14.8 The Galois Group of a Finite Extension of a Finite Field. 233

14.8.1 Proposition . . . . . . . . . 233 14.8.2 The Galois Correspondence 234 14.8.3 Example. . . . . . . . . . . 234 Exercises for Chapter 14 . . . . . . 235 Solutions to Some of the Exercises 243

15 Separable Extensions 15.1 Separability ............. . 15.2 Example of an Inseparable Element. 15.3 A Criterion for Separability ..... 15.4 Perfect Fields . . . . . . . . . . . . . 15.5 Perfect Fields and Separable Extensions 15.6 Galois Extensions.

15.6.1 Definition ......... . 15.6.2 Proposition . . . . . . . . . 15.6.3 The Galois Correspondence Toward Chapter 16 ........ .

16 Recent Developments 16.1 The Inverse Problem of Galois Theory

16.1.1 The Problem ...

257 257 258 258 259 259 260 260 260 260 260

261 261 261

16.1.2 The Abelian Case ....... 262 16.1.3 Example. . . . . . . . . . . . . 262

16.2 Computation of Galois Groups over Q for Small-Degree Poly-nomials . . . . . . . . . . . . . . . . 262 16.2.1 Simplification of the Problem 263 16.2.2 The Irreducibility Problem . 263

xiv Contents

16.2.3 Embedding of G into 8n . . . . . . . . . . . . . . . . 263 16.2.4 Looking for G Among the Transitive Subgroups of 8n 264 16.2.5 Transitive Subgroups of 84 .•.......•.• 264 16.2.6 Study of ~(G) c An . . . . . . . . . . 265 16.2.7 Study of ~(G) c D4 . . . 266 16.2.8 Study of ~(G) c Zj 4Z 267 16.2.9 An Algorithm for n = 4 268

Bibliography 271

Index 277