Galois' Theory Of Algebraic Equations

by Jean-pierre Tignol

Overview

New Edition available hereGalois' Theory of Algebraic Equations 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 main emphasis is placed on equations of at least the third degree, i.e. on the developments during the period from the sixteenth to 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 on the fundamental theorems of modern Galois theory is included. Complete proofs of the quoted results are provided, but the material has been organized in such a way that the most technical details can be skipped by readers who are interested primarily in a broad survey of the theory.This book will appeal to both undergraduate and graduate students in mathematics and the history of science, and also to teachers and mathematicians who wish to obtain a historical perspective of the field. The text has been designed to be self-contained, but some familiarity with basic mathematical structures and with some elementary notions of linear algebra is desirable for a good understanding of the technical discussions in the later chapters.

Product Details

ISBN-13:	9789810245412
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	04/10/2001
Edition description:	New Edition
Pages:	348
Product dimensions:	5.90(w) x 8.40(h) x 0.80(d)

Table of Contents

	Preface	vii
Chapter 1	Quadratic Equations	1
1.1	Introduction	1
1.2	Babylonian algebra	2
1.3	Greek algebra	5
1.4	Arabic algebra	9
Chapter 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 from Cardano's formula	16
Chapter 3	Quartic Equations	21
3.1	The unnaturalness of quartic equations	21
3.2	Ferrari's method	22
Chapter 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
Chapter 5	A Modern Approach to Polynomials	41
5.1	Definitions	41
5.2	Euclidean division	43
5.3	Irreducible polynomials	48
5.4	Roots	50
5.5	Multiple roots and derivatives	53
5.6	Common roots of two polynomials	56
Appendix	Decomposition of rational fractions in sums of partial fractions	58
Chapter 6	Alternative Methods for Cubic and Quartic Equations	61
6.1	Viete on cubic equations	61
6.1.1	Trigonometric solution for the irreducible case	61
6.1.2	Algebraic solution for the general case	62
6.2	Descartes on quartic equations	64
6.3	Rational solutions for equations with rational coefficients	65
6.4	Tschirnhaus' method	67
Chapter 7	Roots of Unity	73
7.1	Introduction	73
7.2	The origin of de Moivre's formula	74
7.3	The roots of unity	81
7.4	Primitive roots and cyclotomic polynomials	86
Appendix	Leibniz and Newton on the summation of series	92
	Exercises	94
Chapter 8	Symmetric Functions	97
8.1	Introduction	97
8.2	Waring's method	100
8.3	The discriminant	106
Appendix	Euler's summation of the series of reciprocals of perfect squares	110
	Exercises	112
Chapter 9	The Fundamental Theorem of Algebra	115
9.1	Introduction	115
9.2	Girard's theorem	116
9.3	Proof of the fundamental theorem	119
Chapter 10	Lagrange	123
10.1	The theory of equations comes of age	123
10.2	Lagrange's observations on previously known methods	127
10.3	First results of group theory and Galois theory	138
	Exercises	150
Chapter 11	Vandermonde	153
11.1	Introduction	153
11.2	The solution of general equations	154
11.3	Cyclotomic equations	158
	Exercises	164
Chapter 12	Gauss on Cyclotomic Equations	167
12.1	Introduction	167
12.2	Number-theoretic preliminaries	168
12.3	Irreducibility of the cyclotomic polynomials of prime index	175
12.4	The periods of cyclotomic equations	182
12.5	Solvability by radicals	192
12.6	Irreducibility of the cyclotomic polynomials	196
Appendix	Ruler and compass construction of regular polygons	200
	Exercises	206
Chapter 13	Ruffini and Abel on General Equations	209
13.1	Introduction	209
13.2	Radical extensions	212
13.3	Abel's theorem on natural irrationalities	218
13.4	Proof of the unsolvability of general equations of degree higher than 4	225
	Exercises	227
Chapter 14	Galois	231
14.1	Introduction	231
14.2	The Galois group of an equation	235
14.3	The Galois group under field extension	254
14.4	Solvability by radicals	264
14.5	Applications	281
Appendix	Galois' description of groups of permutations	295
	Exercises	301
Chapter 15	Epilogue	303
Appendix	The fundamental theorem of Galois theory	307
	Exercises	315
	Selected Solutions	317
	Bibliography	325
	Index	331