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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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