Galois' 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.
Readership: Upper level undergraduates, graduate students and mathematicians in algebra.
Author(s): Jean-Pierre Tignol
Publisher: World Scientific
Year: 2001
Language: English
Commentary: Covers (Front & Back), Optimised
Pages: xiv+333
Chapter 1 Quadratic Equations
1.1 Introduction
1.2 Babylonian algebra
1.3 Greek algebra
1.4 Arabic algebra
Chapter 2 Cubic Equations
2.1 Priority disputes on the solution of cubic equations
2.2 Cardano's formula
2.3 Developments arising from Cardano's formula
Chapter 3 Quartic Equations
3.1 The unnaturalness of quartic equations
3.2 Ferrari's method
Chapter 4 The Creation of Polynomials
4.1 The rise of symbolic algebra
4.1.1 L'Arilhmel que
4.1.2 In A rtem A naiyticemt tsagoge
4.2 Relations between roots and coefficients
Chapter 5 A Modern Approach to Polynomials
5.1 Definitions
5.2 Euclidean division
5.3 Irreducible polynomials
5.4 Roots
5.5 Multiple roots and derivatives
5.6 Common roots of two polynomials
Appendix: Decomposition of rational fractions in sums of partial fractions
Chapter 6 Alternative Methods for Cubic and Quartic Equations
6.1 Viete on cubic equations
6.1.1 Trigonometric solution for the irreducible case
6.1.2 Algebraic solution for the general case
6.2 Descartes on quartic equations
6.3 Rational solutions for equations with rational coefficients
6.4 Tschirnhaus' method
Chapter 7 Roots of Unity
7.1 Introduction
7.2 The origin of de Moiv re's formula
7.3 The roots of unity
7.4 Primitive roots and cyclotomic polynomials
Appendix: Leibniz and Newton on the summation of series
Exercises
Chapter 8 Symmetric Functions
8.1 Introduction
8.2 Waring's method
8.3 The discriminant
Appendix.- Euler's summation of the series of reciprocals of perfect squares
Exercises
Chapter 9 The Fundamental Theorem of Algebra
9.1 Introduction
9.2 Girard?. theorem
9.3 Proof of the fundamental theorem
Chapter 10 Lagrange
10.1 The theory of equations comes of age
10.2 Lagrange's observations on previously known methods
10.3 First results of group theory and Galois theory
Exercises
Chapter 11 Vandermonde
11.1 Introduction
11.2 The solution of general equations
11.3 Cyclotomic equations
Exercises
Chapter 12 Gauss on Cyclotomic Equations
12.1 Introduction
12.2 Number-theoretic Preliminaries
12.3 Irreducibility of the cyclotomic polynomials of prime index
12.4 The periods of cyclotomic equations
12.5 Solvability by radicals
12.6 Irreducibility of the cyclotomic polynomials
Appendix: Ruler and compass construction of regular polygons
Exercises
Chapter 13 Ruffini and Abel on General Equations
13.1 Introduction
13.2 Radical extensions
13.3 Abel's theorem on natural irrationalities
13.4 Proof of the unsolvability of general equations of degree higher than 4
Exercises
Chapter 14 Galois
14.1 Introduction
14.2 The Galois group of an equation
14.3 The Galois group under field extension
14.4 Solvability by radicals
14.5 Applications
Appendix: C alois' description of groups of permutations
Exercises
Chapter 15 Epilogue
Exercises
Selected Solutions
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Bibliography
Index
