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Classical Algebra: Its Nature, Origins, and Uses

ISBN: 978-0-470-27797-3
192 pages
May 2008
Classical Algebra: Its Nature, Origins, and Uses (0470277971) cover image
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.

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Part 1. Numbers and Equations.

Lesson 1. What Algebra Is.

1. Numbers in disguise.

1.1. Classical and modern algebra.

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.

1.1. Geometric interpretations.

2. The classification of equations.

2.1. Diophantine equations.

3. Numerical and formulaic approaches to equations.

3.1. The numerical approach.

3.2. The formulaic approach.

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.

2.1. A linear problem in three unknowns.

3. Systems of linear equations.

4. Polynomial equations.

4.1. Noninteger solutions.

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.

3.1. Cube roots of complex numbers.

4. Alternative forms of the cubic formula.

5. The \irreducible case.

5.1. Imaginary numbers.

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.

3.1. Comparison of the Viète and Cardano solutions.

4. The Tschirnhaus solution of the cubic equation.

5. Lagrange's reflections on the cubic equation.

5.1. The cubic formula in terms of the roots.

5.2. A test case: The quartic.

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.

2.1. A resolvent for the cubic equation.

3. A resolvent for the general quartic equation.

4. The state of polynomial algebra in 1770.

4.1. Seeking a resolvent for the quintic.

5. Permutations enter algebra.

6. Permutations of the variables in a function.

6.1. Two-valued functions.

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.

2.1. The quadratic formula.

2.2. The cubic formula.

2.3. Algebraic functions and algebraic formulas.

3. Abel's proof.

3.1. Taking the formula apart.

3.2. The last step in the proof.

3.3. The verdict on 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.

2.1. Subgroups and cosets.

2.2. Normal subgroups and quotient groups.

2.3. Further analysis of the cubic equation.

2.4. Why the cubic formula must have the form it does.

2.5. Why the roots of unity are important.

2.6. The birth of Galois theory.

3. A sketch of Galois theory.

4. Solution by radicals.

4.1. Abel's theorem.

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.

2.1. Associative rings.

2.2. Lie rings.

2.3. Special classes of rings.

3. Division rings and fields.

4. Vector spaces and related structures.

4.1. Modules.

4.2. Algebras.

5. Conclusion.

Appendix: Some Facts about Polynomials.

Answers to the Problems and Questions.

Subject Index.

Name Index.

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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).
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  • Structured as eleven "lessons" that are intended to give the reader perspective on algebra as well as explain what it is about, where it came from, and why it is important

  • Addresses prominent features of algebra (i.e. Equations and Their Solutions, Numerical Solutions of Equations, The Systematic Study of Equations, Galois Theory, in an effort to tie together the disconnect between algebra and modern algebra

  • Provides a mixture of historical remarks and analysis of polynomial equations throughout the "lessons"

  • Intended as a popularization of the subject of algebra for the edification of students and interested readers, and simultaneously acts as a pedagogical aid for teachers who may use it to provide context and perspective on the subject for their students

  • Contains thought-provoking questions and problems in algebra, for which complete answers are provided in an appendix.

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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)


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