Classical Algebra : Its Nature, Origins, and Uses

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  • Format: Paperback
  • Copyright: 2008-03-07
  • Publisher: Wiley-Interscience

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Supplemental Materials

What is included with this book?


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.

Author Biography

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).

Table of Contents

Prefacep. ix
Numbers and Equationsp. 1
What Algebra Isp. 3
Numbers in disguisep. 3
"Classical" and modern algebrap. 5
Arithmetic and algebrap. 7
The "environment" of algebra: Number systemsp. 8
Important concepts and principles in this lessonp. 11
Problems and questionsp. 12
Further readingp. 15
Equations and Their Solutionsp. 17
Polynomial equations, coefficients, and rootsp. 17
Geometric interpretationsp. 18
The classification of equationsp. 19
Diophantine equationsp. 20
Numerical and formulaic approaches to equationsp. 20
The numerical approachp. 21
The formulaic approachp. 21
Important concepts and principles in this lessonp. 23
Problems and questionsp. 23
Further readingp. 24
Where Algebra Comes Fromp. 25
An Egyptian problemp. 25
A Mesopotamian problemp. 26
A Chinese problemp. 26
An Arabic problemp. 27
A Japanese problemp. 28
Problems and questionsp. 29
Further readingp. 30
Why Algebra Is Importantp. 33
Example: An ideal pendulump. 35
Problems and questionsp. 38
Further readingp. 44
Numerical Solution of Equationsp. 45
A simple but crude methodp. 45
Ancient Chinese methods of calculatingp. 46
A linear problem in three unknownsp. 47
Systems of linear equationsp. 48
Polynomial equationsp. 49
Noninteger solutionsp. 50
The cubic equationp. 51
Problems and questionsp. 52
Further readingp. 53
The Formulaic Approach to Equationsp. 55
Combinatoric Solutions I: Quadratic Equationsp. 57
Why not set up tables of solutions?p. 57
The quadratic formulap. 60
Problems and questionsp. 61
Further readingp. 62
Combinatoric Solutions II: Cubic Equationsp. 63
Reduction from four parameters to onep. 63
Graphical solutions of cubic equationsp. 64
Efforts to find a cubic formulap. 65
Cube roots of complex numbersp. 67
Alternative forms of the cubic formulap. 68
The "irreducible case"p. 69
Imaginary numbersp. 70
Problems and questionsp. 71
Further readingp. 72
Resolventsp. 73
From Combinatorics to Resolventsp. 75
Solution of the irreducible case using complex numbersp. 76
The quartic equationp. 77
Viete's solution of the irreducible case of the cubicp. 78
Comparison of the Viete and Cardano solutionsp. 79
The Tschirnhaus solution of the cubic equationp. 80
Lagrange's reflections on the cubic equationp. 82
The cubic formula in terms of the rootsp. 83
A test case: The quarticp. 84
Problems and questionsp. 85
Further readingp. 88
The Search for Resolventsp. 91
Coefficients and rootsp. 92
A unified approach to equations of all degreesp. 92
A resolvent for the cubic equationp. 93
A resolvent for the general quartic equationp. 93
The state of polynomial algebra in 1770p. 95
Seeking a resolvent for the quinticp. 97
Permutations enter algebrap. 98
Permutations of the variables in a functionp. 98
Two-valued functionsp. 100
Problems and questionsp. 101
Further readingp. 105
Abstract Algebrap. 107
Existence and Constructibility of Rootsp. 109
Proof that the complex numbers are algebraically closedp. 109
Solution by radicals: General considerationsp. 112
The quadratic formulap. 112
The cubic formulap. 116
Algebraic functions and algebraic formulasp. 118
Abel's proofp. 119
Taking the formula apartp. 120
The last step in the proofp. 121
The verdict on Abel's proofp. 121
Problems and questionsp. 122
Further readingp. 122
The Breakthrough: Galois Theoryp. 125
An example of a solving an equation by radicalsp. 126
Field automorphisms and permutations of rootsp. 127
Subgroups and cosetsp. 129
Normal subgroups and quotient groupsp. 129
Further analysis of the cubic equationp. 130
Why the cubic formula must have the form it doesp. 131
Why the roots of unity are importantp. 132
The birth of Galois theoryp. 133
A sketch of Galois theoryp. 135
Solution by radicalsp. 136
Abel's theoremp. 137
Some simple examples for practicep. 138
The story of polynomial algebra: a recapp. 146
Problems and questionsp. 147
Further readingp. 149
Epilogue: Modern Algebrap. 151
Groupsp. 151
Ringsp. 154
Associative ringsp. 154
Lie ringsp. 155
Special classes of ringsp. 156
Division rings and fieldsp. 156
Vector spaces and related structuresp. 156
Modulesp. 157
Algebrasp. 158
Conclusionp. 158
Some Facts about Polynomialsp. 161
Answers to the Problems and Questionsp. 167
Subject Indexp. 197
Name Indexp. 205
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