|  | Cooke, Roger M.
Classical Algebra: Its Nature, Origins, and Uses
 1. Edition - April 2008
53.90 Euro
2008. 224 Pages, Softcover
ISBN-10: 0-470-25952-3
ISBN-13: 978-0-470-25952-8 - John Wiley & Sons
Sample Chapter
Short description
This book combines history, pedagogy, and popularization in a unique way to produce a unified and coherent picture of algebra. It is an excellent source for readers to grasp the essence of algebra as a whole, including how it has developed and what it has developed into. The book answers questions such as: 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, and how were those problems solved and those issues resolved? Most of these questions do not get answered, or even raised, in courses in algebra and those that do get raised in algebra courses are generally neglected in courses in the history of mathematics.
From the contents
Preface
Part 1. Numbers and Equations.
Lesson 1. What Algebra Is.
1. Numbers in disguise.
2. Arithmetic and algebra.
3. The environment of algebra: Number systems.
4. Important concepts and principles in this lesson.
5. Problems and questions.
6. Further reading.
Lesson 2. Equations and Their Solutions.
1. Polynomial equations, coefficients, and roots.
2. The classification of equations.
3. Numerical and formulaic approaches to equations.
4. Important concepts and principles in this lesson.
5. Problems and questions.
6. Further reading.
Lesson 3. Where Algebra Comes From.
1. An Egyptian problem.
2. A Mesopotamian problem.
3. A Chinese problem.
4. An Arabic problem.
5. A Japanese problem.
6. Problems and questions.
7. Further reading.
Lesson 4. Why Algebra Is Important.
1. Example: An ideal pendulum.
2. Problems and questions.
3. Further reading.
Lesson 5. Numerical Solution of Equations.
1. A simple but crude method.
2. Ancient Chinese methods of calculating.
3. Systems of linear equations.
4. Polynomial equations.
5. The cubic equation.
6. Problems and questions.
7. Further reading.
Part 2. The Formulaic Approach to Equations.
Lesson 6. Combinatoric Solutions I: Quadratic Equations.
1. Why not set up tables of solutions?.
2. The quadratic formula.
3. Problems and questions.
4. Further reading.
Lesson 7. Combinatoric Solutions II: Cubic Equations.
1. Reduction from four parameters to one.
2. Graphical solutions of cubic equations.
3. Efforts to find a cubic formula.
4. Alternative forms of the cubic formula.
5. The \irreducible case.
6. Problems and questions.
7. Further reading.
Part 3. Resolvents.
Lesson 8. From Combinatorics to Resolvents.
1. Solution of the irreducible case using complex numbers.
2. The quartic equation.
3. Viμete's solution of the irreducible case of the cubic.
4. The Tschirnhaus solution of the cubic equation.
5. Lagrange's reflections on the cubic equation.
6. Problems and questions.
7. Further reading.
Lesson 9. The Search for Resolvents.
1. Coefficients and roots.
2. A unified approach to equations of all degrees.
3. A resolvent for the general quartic equation.
4. The state of polynomial algebra in 1770.
5. Permutations enter algebra.
6. Permutations of the variables in a function.
7. Problems and questions.
8. Further reading.
Part 4. Abstract Algebra.
Lesson 10. Existence and Constructibility of Roots.
1. Proof that the complex numbers are algebraically closed.
2. Solution by radicals: General considerations.
3. Abel's proof.
4. Problems and questions.
5. Further reading.
Lesson 11. The Breakthrough: Galois Theory.
1. An example of a solving an equation by radicals.
2. Field automorphisms and permutations of roots.
3. A sketch of Galois theory.
4. Solution by radicals.
5. Some simple examples for practice.
6. The story of polynomial algebra: a recap.
7. Problems and questions.
8. Further reading.
Epilogue: Modern Algebra.
1. Groups.
2. Rings.
3. Division rings and fields.
4. Vector spaces and related structures.
5. Conclusion.
Appendix: Some Facts about Polynomials.
Answers to the Problems and Questions.
Subject Index.
Name Index.
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