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.
Rezensionen / Stimmen
Winner of the 2008 CHOICE Outstanding Academic Title Award "The book is written in an utmost lucid and cultured style, basically accessible to anyone with a general interest in mathematics, and a valuable contribution to the overall popularization of algebra into the bargain." (Zentralblatt Math, 2008)
"Cooke does a superb job of beginning to bridge the gulf between quadratic equations and group theory." (CHOICE, September 2008)
"Classical Algebra deserves a large audience, and should be in every undergraduate library." (MAA Reviews, June 2, 2008)
Produkt-Info
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Sprache
Verlagsort
Verlagsgruppe
Zielgruppe
Produkt-Hinweis
Broschur/Paperback
Klebebindung
Illustrationen
Drawings: 3 B&W, 0 Color; Graphs: 11 B&W, 0 Color
Maße
Höhe: 234 mm
Breite: 156 mm
Dicke: 12 mm
Gewicht
ISBN-13
978-0-470-25952-8 (9780470259528)
Schweitzer Klassifikation
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).
Autor*in
Univ. of Vermont
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.
