Cover image for Classical algebra : its nature, origins, and uses
Title:
Classical algebra : its nature, origins, and uses
Personal Author:
Publication Information:
Haboken, NJ : Wiley-Interscience, 2008
Physical Description:
xii, 206 p. : ill. ; 24 cm.
ISBN:
9780470259528

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30000010178233 QA155 C664 2008 Open Access Book Book
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Summary

Summary

This insightful book combines the history, pedagogy, and popularization of algebra to present a unified discussion of the subject.

Classical Algebra 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, this book facilitates a thorough understanding of algebra and illustrates how the concepts of modern algebra originally developed from classical algebraic precursors.

This book successfully ties together the disconnect between classical and modern algebraand provides readers with answers to many fascinating questions that typically go unexamined, including:

What is algebra about?

How did it arise?

What uses does it have?

How did it develop?

What problems and issues have occurred in its history?

How were these problems and issues resolved?

The author answers these questions and more, shedding light on a rich history of the subject--from ancient and medieval times to the present. Structured as eleven "lessons" that are intended to give the reader further insight on classical algebra, each chapter contains thought-provoking problems and stimulating questions, for which complete answers are provided in an appendix.

Complemented with a mixture of historical remarks and analyses of polynomial equations throughout, Classical Algebra: Its Nature, Origins, and Uses is an excellent book for mathematics courses at the undergraduate level. It also serves as a valuable resource to anyone with a general interest in mathematics.


Author Notes

Roger Cooke is Emeritus Professor of Mathematics in the Department of Mathematics and Statistics at the University of Vermont.


Reviews 1

Choice Review

The development of (polynomial) algebra from ancient times to the present is the emphasis of this book. Cooke (emer., Univ. of Vermont) attempts to fill the gap between the algebra taught in the secondary curriculum, and the abstract algebra typically studied as an undergraduate. At the pre-undergraduate level, the goals of algebra are primarily to find roots of polynomials, and to solve rational equations and systems of linear equations. In undergraduate algebra courses, students are typically exposed to the abstract concepts of groups, rings, fields, and perhaps some Galois theory. Although the length of a book to fill all the gaps would be prohibitive, Cooke gives valuable insight into the historical hows and whys of abstract algebra. He uses his expertise in the history of 20th-century physics to motivate uses of algebra through numerous examples, including Kepler's third law, Coulomb's law, and magnetic induction. The general solutions to quadratics, cubics, and quartics are developed and presented in historical context. The reader will find the more abstract mathematics explained through carefully worked-out examples in full detail. Cooke does a superb job of beginning to bridge the gulf between quadratic equations and group theory. Summing Up: Recommended. Upper-division undergraduates and researchers/faculty. J. T. Zerger Catawba College


Table of Contents

Prefacep. ix
Part 1 Numbers and Equationsp. 1
Lesson 1 What Algebra Isp. 3
1 Numbers in disguisep. 3
1.1 "Classical" and modern algebrap. 5
2 Arithmetic and algebrap. 7
3 The "environment" of algebra: Number systemsp. 8
4 Important concepts and principles in this lessonp. 11
5 Problems and questionsp. 12
6 Further readingp. 15
Lesson 2 Equations and Their Solutionsp. 17
1 Polynomial equations, coefficients, and rootsp. 17
1.1 Geometric interpretationsp. 18
2 The classification of equationsp. 19
2.1 Diophantine equationsp. 20
3 Numerical and formulaic approaches to equationsp. 20
3.1 The numerical approachp. 21
3.2 The formulaic approachp. 21
4 Important concepts and principles in this lessonp. 23
5 Problems and questionsp. 23
6 Further readingp. 24
Lesson 3 Where Algebra Comes Fromp. 25
1 An Egyptian problemp. 25
2 A Mesopotamian problemp. 26
3 A Chinese problemp. 26
4 An Arabic problemp. 27
5 A Japanese problemp. 28
6 Problems and questionsp. 29
7 Further readingp. 30
Lesson 4 Why Algebra Is Importantp. 33
1 Example: An ideal pendulump. 35
2 Problems and questionsp. 38
3 Further readingp. 44
Lesson 5 Numerical Solution of Equationsp. 45
1 A simple but crude methodp. 45
2 Ancient Chinese methods of calculatingp. 46
2.1 A linear problem in three unknownsp. 47
3 Systems of linear equationsp. 48
4 Polynomial equationsp. 49
4.1 Noninteger solutionsp. 50
5 The cubic equationp. 51
6 Problems and questionsp. 52
7 Further readingp. 53
Part 2 The Formulaic Approach to Equationsp. 55
Lesson 6 Combinatoric Solutions I: Quadratic Equationsp. 57
1 Why not set up tables of solutions?p. 57
2 The quadratic formulap. 60
3 Problems and questionsp. 61
4 Further readingp. 62
Lesson 7 Combinatoric Solutions II: Cubic Equationsp. 63
1 Reduction from four parameters to onep. 63
2 Graphical solutions of cubic equationsp. 64
3 Efforts to find a cubic formulap. 65
3.1 Cube roots of complex numbersp. 67
4 Alternative forms of the cubic formulap. 68
5 The "irreducible case"p. 69
5.1 Imaginary numbersp. 70
6 Problems and questionsp. 71
7 Further readingp. 72
Part 3 Resolventsp. 73
Lesson 8 From Combinatorics to Resolventsp. 75
1 Solution of the irreducible case using complex numbersp. 76
2 The quartic equationp. 77
3 Viete's solution of the irreducible case of the cubicp. 78
3.1 Comparison of the Viete and Cardano solutionsp. 79
4 The Tschirnhaus solution of the cubic equationp. 80
5 Lagrange's reflections on the cubic equationp. 82
5.1 The cubic formula in terms of the rootsp. 83
5.2 A test case: The quarticp. 84
6 Problems and questionsp. 85
7 Further readingp. 88
Lesson 9 The Search for Resolventsp. 91
1 Coefficients and rootsp. 92
2 A unified approach to equations of all degreesp. 92
2.1 A resolvent for the cubic equationp. 93
3 A resolvent for the general quartic equationp. 93
4 The state of polynomial algebra in 1770p. 95
4.1 Seeking a resolvent for the quinticp. 97
5 Permutations enter algebrap. 98
6 Permutations of the variables in a functionp. 98
6.1 Two-valued functionsp. 100
7 Problems and questionsp. 101
8 Further readingp. 105
Part 4 Abstract Algebrap. 107
Lesson 10 Existence and Constructibility of Rootsp. 109
1 Proof that the complex numbers are algebraically closedp. 109
2 Solution by radicals: General considerationsp. 112
2.1 The quadratic formulap. 112
2.2 The cubic formulap. 116
2.3 Algebraic functions and algebraic formulasp. 118
3 Abel's proofp. 119
3.1 Taking the formula apartp. 120
3.2 The last step in the proofp. 121
3.3 The verdict on Abel's proofp. 121
4 Problems and questionsp. 122
5 Further readingp. 122
Lesson 11 The Breakthrough: Galois Theoryp. 125
1 An example of a solving an equation by radicalsp. 126
2 Field automorphisms and permutations of rootsp. 127
2.1 Subgroups and cosetsp. 129
2.2 Normal subgroups and quotient groupsp. 129
2.3 Further analysis of the cubic equationp. 130
2.4 Why the cubic formula must have the form it doesp. 131
2.5 Why the roots of unity are importantp. 132
2.6 The birth of Galois theoryp. 133
3 A sketch of Galois theoryp. 135
4 Solution by radicalsp. 136
4.1 Abel's theoremp. 137
5 Some simple examples for practicep. 138
6 The story of polynomial algebra: a recapp. 146
7 Problems and questionsp. 147
8 Further readingp. 149
Epilogue: Modern Algebrap. 151
1 Groupsp. 151
2 Ringsp. 154
2.1 Associative ringsp. 154
2.2 Lie ringsp. 155
2.3 Special classes of ringsp. 156
3 Division rings and fieldsp. 156
4 Vector spaces and related structuresp. 156
4.1 Modulesp. 157
4.2 Algebrasp. 158
5 Conclusionp. 158
Appendix Some Facts about Polynomialsp. 161
Answers to the Problems and Questionsp. 167
Subject Indexp. 197
Name Indexp. 205