Abstract Algebra and Famous Impossibilities

1. Front Matter

   Pages i-x

   PDF

2. Introduction

   • Arthur Jones, Kenneth R. Pearson, Sidney A. Morris

   Pages 1-5

3. Algebraic Preliminaries

   • Arthur Jones, Kenneth R. Pearson, Sidney A. Morris

   Pages 7-26

4. Algebraic Numbers and Their Polynomials

   • Arthur Jones, Kenneth R. Pearson, Sidney A. Morris

   Pages 27-37

5. Extending Fields

   • Arthur Jones, Kenneth R. Pearson, Sidney A. Morris

   Pages 39-59

6. Irreducible Polynomials

   • Arthur Jones, Kenneth R. Pearson, Sidney A. Morris

   Pages 61-74

7. Straightedge and Compass Constructions

   • Arthur Jones, Kenneth R. Pearson, Sidney A. Morris

   Pages 75-98

8. Proofs of the Impossibilities

   • Arthur Jones, Kenneth R. Pearson, Sidney A. Morris

   Pages 99-113

9. Transcendence of e and π

   • Arthur Jones, Kenneth R. Pearson, Sidney A. Morris

   Pages 115-161

10. An Algebraic Postscript

    • Arthur Jones, Kenneth R. Pearson, Sidney A. Morris

    Pages 163-175

11. Other Impossibilities and Abstract Algebra

    • Arthur Jones, Kenneth R. Pearson, Sidney A. Morris

    Pages 177-182

12. Back Matter

    Pages 183-189

    PDF

The famous problems of squaring the circle, doubling the cube and trisecting an angle captured the imagination of both professional and amateur mathematicians for over two thousand years. Despite the enormous effort and ingenious attempts by these men and women, the problems would not yield to purely geometrical methods. It was only the development. of abstract algebra in the nineteenth century which enabled mathematicians to arrive at the surprising conclusion that these constructions are not possible. In this book we develop enough abstract algebra to prove that these constructions are impossible. Our approach introduces all the relevant concepts about fields in a way which is more concrete than usual and which avoids the use of quotient structures (and even of the Euclidean algorithm for finding the greatest common divisor of two polynomials). Having the geometrical questions as a specific goal provides motivation for the introduction of the algebraic concepts and we have found that students respond very favourably. We have used this text to teach second-year students at La Trobe University over a period of many years, each time refining the material in the light of student performance.

• Finite
• Irreducibility
• PostScript
• Vector space
• algebra
• algorithms
• boundary element method
• construction
• form
• integral
• integration
• polynomial
• proof
• proving
• ring

• Department of Mathematics, La Trobe University, Bundoora, Australia

  Arthur Jones, Kenneth R. Pearson

• Faculty of Informatics, University of Wollongong, Wollongong, Australia

  Sidney A. Morris

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