دانلود کتاب Galois theory of algebraic equations

فهرست مطالب :

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