Classical Algebra Its Nature, Origins, and

This book provides a complete and contemporary perspective on classical polynomial algebra through the exploration of how it was developed and how it exists today. With a focus on prominent areas such as the numerical solutions of equations, the systematic study of equations, and Galois theory, the book facilitates a thorough understanding of algebra and illustrates how the concepts of modern algebra originally developed from classical algebraic precursors. It is complemented with historical remarks and analyses of polynomial equations throughout.

Roger Cooke, PhD, is Emeritus Professor of Mathematics in the Department of Mathematics and Statistics at the University of Vermont. Dr. Cooke has over forty years of academic experience, and his areas of research interest include the history of mathematics, almost-periodic functions, uniqueness of trigonometric series representations, and Fourier analysis. He is also the author of The History of Mathematics: A Brief Course, Second Edition (Wiley).

Preface	p. ix
Numbers and Equations	p. 1
What Algebra Is	p. 3
Numbers in disguise	p. 3
"Classical" and modern algebra	p. 5
Arithmetic and algebra	p. 7
The "environment" of algebra: Number systems	p. 8
Important concepts and principles in this lesson	p. 11
Problems and questions	p. 12
Further reading	p. 15
Equations and Their Solutions	p. 17
Polynomial equations, coefficients, and roots	p. 17
Geometric interpretations	p. 18
The classification of equations	p. 19
Diophantine equations	p. 20
Numerical and formulaic approaches to equations	p. 20
The numerical approach	p. 21
The formulaic approach	p. 21
Important concepts and principles in this lesson	p. 23
Problems and questions	p. 23
Further reading	p. 24
Where Algebra Comes From	p. 25
An Egyptian problem	p. 25
A Mesopotamian problem	p. 26
A Chinese problem	p. 26
An Arabic problem	p. 27
A Japanese problem	p. 28
Problems and questions	p. 29
Further reading	p. 30
Why Algebra Is Important	p. 33
Example: An ideal pendulum	p. 35
Problems and questions	p. 38
Further reading	p. 44
Numerical Solution of Equations	p. 45
A simple but crude method	p. 45
Ancient Chinese methods of calculating	p. 46
A linear problem in three unknowns	p. 47
Systems of linear equations	p. 48
Polynomial equations	p. 49
Noninteger solutions	p. 50
The cubic equation	p. 51
Problems and questions	p. 52
Further reading	p. 53
The Formulaic Approach to Equations	p. 55
Combinatoric Solutions I: Quadratic Equations	p. 57
Why not set up tables of solutions?	p. 57
The quadratic formula	p. 60
Problems and questions	p. 61
Further reading	p. 62
Combinatoric Solutions II: Cubic Equations	p. 63
Reduction from four parameters to one	p. 63
Graphical solutions of cubic equations	p. 64
Efforts to find a cubic formula	p. 65
Cube roots of complex numbers	p. 67
Alternative forms of the cubic formula	p. 68
The "irreducible case"	p. 69
Imaginary numbers	p. 70
Problems and questions	p. 71
Further reading	p. 72
Resolvents	p. 73
From Combinatorics to Resolvents	p. 75
Solution of the irreducible case using complex numbers	p. 76
The quartic equation	p. 77
Viete's solution of the irreducible case of the cubic	p. 78
Comparison of the Viete and Cardano solutions	p. 79
The Tschirnhaus solution of the cubic equation	p. 80
Lagrange's reflections on the cubic equation	p. 82
The cubic formula in terms of the roots	p. 83
A test case: The quartic	p. 84
Problems and questions	p. 85
Further reading	p. 88
The Search for Resolvents	p. 91
Coefficients and roots	p. 92
A unified approach to equations of all degrees	p. 92
A resolvent for the cubic equation	p. 93
A resolvent for the general quartic equation	p. 93
The state of polynomial algebra in 1770	p. 95
Seeking a resolvent for the quintic	p. 97
Permutations enter algebra	p. 98
Permutations of the variables in a function	p. 98
Two-valued functions	p. 100
Problems and questions	p. 101
Further reading	p. 105
Abstract Algebra	p. 107
Existence and Constructibility of Roots	p. 109
Proof that the complex numbers are algebraically closed	p. 109
Solution by radicals: General considerations	p. 112
The quadratic formula	p. 112
The cubic formula	p. 116
Algebraic functions and algebraic formulas	p. 118
Abel's proof	p. 119
Taking the formula apart	p. 120
The last step in the proof	p. 121
The verdict on Abel's proof	p. 121
Problems and questions	p. 122
Further reading	p. 122
The Breakthrough: Galois Theory	p. 125
An example of a solving an equation by radicals	p. 126
Field automorphisms and permutations of roots	p. 127
Subgroups and cosets	p. 129
Normal subgroups and quotient groups	p. 129
Further analysis of the cubic equation	p. 130
Why the cubic formula must have the form it does	p. 131
Why the roots of unity are important	p. 132
The birth of Galois theory	p. 133
A sketch of Galois theory	p. 135
Solution by radicals	p. 136
Abel's theorem	p. 137
Some simple examples for practice	p. 138
The story of polynomial algebra: a recap	p. 146
Problems and questions	p. 147
Further reading	p. 149
Epilogue: Modern Algebra	p. 151
Groups	p. 151
Rings	p. 154
Associative rings	p. 154
Lie rings	p. 155
Special classes of rings	p. 156
Division rings and fields	p. 156
Vector spaces and related structures	p. 156
Modules	p. 157
Algebras	p. 158
Conclusion	p. 158
Some Facts about Polynomials	p. 161
Answers to the Problems and Questions	p. 167
Subject Index	p. 197
Name Index	p. 205
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Classical Algebra Its Nature, Origins, and Uses

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