Abstract Algebra and Famous Impossibilities

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

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-x
  2. Arthur Jones, Kenneth R. Pearson, Sidney A. Morris
    Pages 1-5
  3. Arthur Jones, Kenneth R. Pearson, Sidney A. Morris
    Pages 7-26
  4. Arthur Jones, Kenneth R. Pearson, Sidney A. Morris
    Pages 27-37
  5. Arthur Jones, Kenneth R. Pearson, Sidney A. Morris
    Pages 39-59
  6. Arthur Jones, Kenneth R. Pearson, Sidney A. Morris
    Pages 61-74
  7. Arthur Jones, Kenneth R. Pearson, Sidney A. Morris
    Pages 75-98
  8. Arthur Jones, Kenneth R. Pearson, Sidney A. Morris
    Pages 99-113
  9. Arthur Jones, Kenneth R. Pearson, Sidney A. Morris
    Pages 115-161
  10. Arthur Jones, Kenneth R. Pearson, Sidney A. Morris
    Pages 163-175
  11. Arthur Jones, Kenneth R. Pearson, Sidney A. Morris
    Pages 177-182
  12. Back Matter
    Pages 183-189

About this book

Introduction

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.

Keywords

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

Authors and affiliations

  • Arthur Jones
    • 1
  • Kenneth R. Pearson
    • 1
  • Sidney A. Morris
    • 2
  1. 1.Department of MathematicsLa Trobe UniversityBundooraAustralia
  2. 2.Faculty of InformaticsUniversity of WollongongWollongongAustralia

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4419-8552-1
  • Copyright Information Springer-Verlag New York, Inc. 1991
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-97661-7
  • Online ISBN 978-1-4419-8552-1
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • About this book