Galois' Theory Of Algebraic Equations (Second Edition)

by Jean-pierre Tignol

Overview

The book gives a detailed account of the development of the theory of algebraic equations, from its origins in ancient times to its completion by Galois in the nineteenth century. The appropriate parts of works by Cardano, Lagrange, Vandermonde, Gauss, Abel, and Galois are reviewed and placed in their historical perspective, with the aim of conveying to the reader a sense of the way in which the theory of algebraic equations has evolved and has led to such basic mathematical notions as 'group' and 'field'. A brief discussion of the fundamental theorems of modern Galois theory and complete proofs of the quoted results are provided, and the material is organized in such a way that the more technical details can be skipped by readers who are interested primarily in a broad survey of the theory.In this second edition, the exposition has been improved throughout and the chapter on Galois has been entirely rewritten to better reflect Galois' highly innovative contributions. The text now follows more closely Galois' memoir, resorting as sparsely as possible to anachronistic modern notions such as field extensions. The emerging picture is a surprisingly elementary approach to the solvability of equations by radicals, and yet is unexpectedly close to some of the most recent methods of Galois theory.

Product Details

ISBN-13:	9789814704694
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	02/24/2016
Pages:	324
Product dimensions:	6.00(w) x 9.10(h) x 0.90(d)

Table of Contents

Preface to the Second Edition vii

Preface to the First Edition (2001) ix

1 Quadratic Equations 1

1.1 Babylonian algebra 2

1.2 Greek algebra 5

1.3 Arabic algebra 9

2 Cubic Equations 13

2.1 Priority disputes on the solution of cubic equations 13

2.2 Cardano's formula 15

2.3 Developments arising horn Cardano's formula 16

3 Quartic Equations 21

3.1 The unnaturalness of quartic equations 21

3.2 Ferrari's method 22

4 The Creation of Polynomials 25

4.1 The rise of symbolic algebra 25

4.1.1 L'Arithmetique 26

4.1.2 In Artem Analyticem Isagoge 29

4.2 Relations between roots and coefficients 30

5 A Modern Approach to Polynomials 41

5.1 Definitions 41

5.2 Euclidean division 43

5.3 Irreducible polynomials 48

5.4 Roots 51

5.5 Multiple roots and derivatives 53

5.6 Common roots of two polynomials 56

Appendix: Decomposition of rational functions into sums of partial fractions 60

6 Alternative Methods for Cubic and Quartic Equations 63

6.1 Viète on cubic equations 63

6.1.1 Trigonometric solution for the irreducible case 63

6.1.2 Algebraic solution for the general case 64

6.2 Descartes on quartic equations 66

6.3 Rational solutions for equations with rational coefficients 67

6.4 Tschirnhaus' method 68

7 Roots of Unity 73

7.1 The origins of de Moivre's formula 73

7.2 The roots of unity 80

7.3 Primitive roots and cyclotomic polynomials 85

Appendix: Leibniz and Newton on the summation of series 89

Exercises 90

8 Symmetric Functions 93

8.1 Waring's method 96

8.2 The discriminant 101

Appendix: Euler's summation of the series of reciprocals of perfect squares 105

Exercises 107

9 The Fundamental Theorem of Algebra 109

9.1 Girard's theorem 110

9.2 Proof of the fundamental theorem 113

10 Lagrange 117

10.1 The theory of equations comes of age 117

10.2 Lagrange's observations on previously known methods 121

10.3 First results of group theory and Galois theory 131

Exercises 142

11 Vandermonde 143

11.1 The solution of general equations 144

11.2 Cyclotomic equations 148

Exercises 154

12 Gauss on Cyclotomic Equations 155

12.1 Number-theoretic preliminaries 156

12.2 Irreducibility of the cyclotomic polynomials of prime index 162

12.3 The periods of cyclotomic equations 169

12.4 Solvability by radicals 178

12.5 Irreducibility of the cyclotomic polynomials 182

Appendix: Ruler and compass construction of regular polygons 185

Exercises 192

13 Ruffiui and Abel on General Equations 193

13.1 Radical extensions 195

13.2 Abel's theorem on natural irrationalities 203

13.3 Proof of the unsolvability of general equations of degree higher than 4 209

Exercises 211

14 Galois 215

14.1 Arrangements and permutations 220

14.2 The Galois group of an equation 225

14.3 The Galois group under base field extension 236

14.4 Solvability by radicals 246

14.5 Applications 256

14.5.1 Irreducible equations of prime degree 256

14.5.2 Abelian equations 265

Exercises 268

15 Epilogue 271

Appendix 1 The fundamental theorem of Galois theory 274

Appendix 2 Galois theory à la Grothendieck 283

Étale algebras 283

Galois algebras 285

Galois groups 287

Exercises 290

Selected Solutions 291

Bibliography 299

Index 305