A Course in Constructive Algebra

  • Ray Mines
  • Fred Richman
  • Wim Ruitenburg

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 1-34
  3. Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 35-77
  4. Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 78-107
  5. Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 108-127
  6. Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 128-138
  7. Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 139-175
  8. Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 176-192
  9. Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 193-231
  10. Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 232-248
  11. Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 249-264
  12. Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 265-286
  13. Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 287-325
  14. Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 326-334
  15. Back Matter
    Pages 335-344

About this book

Introduction

The constructive approach to mathematics has enjoyed a renaissance, caused in large part by the appearance of Errett Bishop's book Foundations of constr"uctiue analysis in 1967, and by the subtle influences of the proliferation of powerful computers. Bishop demonstrated that pure mathematics can be developed from a constructive point of view while maintaining a continuity with classical terminology and spirit; much more of classical mathematics was preserved than had been thought possible, and no classically false theorems resulted, as had been the case in other constructive schools such as intuitionism and Russian constructivism. The computers created a widespread awareness of the intuitive notion of an effecti ve procedure, and of computation in principle, in addi tion to stimulating the study of constructive algebra for actual implementation, and from the point of view of recursive function theory. In analysis, constructive problems arise instantly because we must start with the real numbers, and there is no finite procedure for deciding whether two given real numbers are equal or not (the real numbers are not discrete) . The main thrust of constructive mathematics was in the direction of analysis, although several mathematicians, including Kronecker and van der waerden, made important contributions to construc­ tive algebra. Heyting, working in intuitionistic algebra, concentrated on issues raised by considering algebraic structures over the real numbers, and so developed a handmaiden'of analysis rather than a theory of discrete algebraic structures.

Keywords

Galois theory algebra field matrices matrix torsion transformation

Authors and affiliations

  • Ray Mines
    • 1
  • Fred Richman
    • 1
  • Wim Ruitenburg
    • 2
  1. 1.Department of Mathematical SciencesNew Mexico State UniversityLas CrucesUSA
  2. 2.Department of Mathematics, Statistics, and Computer ScienceMarquette UniversityMilwaukeeUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4419-8640-5
  • Copyright Information Springer-Verlag New York Inc. 1988
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-96640-3
  • Online ISBN 978-1-4419-8640-5
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • About this book