Preface
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'Arithmetique
4.1.2 In Artem Analyticem Isagoge .
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 Vi~te 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 Moivre'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's 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 Wasndermonde
Chapter 12 Gauss on Cyclotomle Ewquations
Chapter 13 Ruffini and A bel on General Equations
Chapter 14 Galois
Chapter 15 Epilogue
Index
