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A Course in Constructive Algebra

  • Book
  • © 1988

Overview

Authors:
  1. Ray Mines

    1. Department of Mathematical Sciences, New Mexico State University, Las Cruces, USA

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  2. Fred Richman

    1. Department of Mathematical Sciences, New Mexico State University, Las Cruces, USA

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  3. Wim Ruitenburg

    1. Department of Mathematics, Statistics, and Computer Science, Marquette University, Milwaukee, USA

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Part of the book series: Universitext (UTX)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xi
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  2. Sets

    • Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 1-34

  3. Basic Algebra

    • Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 35-77

  4. Rings and Modules

    • Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 78-107

  5. Divisibility in Discrete Domains

    • Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 108-127

  6. Principal Ideal Domains

    • Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 128-138

  7. Field Theory

    • Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 139-175

  8. Factoring Polynomials

    • Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 176-192

  9. Commutative Noetherian Rings

    • Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 193-231

  10. Finite Dimensional Algebras

    • Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 232-248

  11. Free Groups

    • Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 249-264

  12. Abelian Groups

    • Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 265-286

  13. Valuation Theory

    • Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 287-325

  14. Dedekind Domains

    • Ray Mines, Fred Richman, Wim Ruitenburg
    Pages 326-334

  15. Back Matter

    Pages 335-344
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematical Sciences, New Mexico State University, Las Cruces, USA

    Ray Mines, Fred Richman

  • Department of Mathematics, Statistics, and Computer Science, Marquette University, Milwaukee, USA

    Wim Ruitenburg

Bibliographic Information

  • Book Title: A Course in Constructive Algebra

  • Authors: Ray Mines, Fred Richman, Wim Ruitenburg

  • Series Title: Universitext

  • DOI: https://doi.org/10.1007/978-1-4419-8640-5

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1988

  • Softcover ISBN: 978-0-387-96640-3Published: 18 December 1987

  • eBook ISBN: 978-1-4419-8640-5Published: 10 September 2012

  • Series ISSN: 0172-5939

  • Series E-ISSN: 2191-6675

  • Edition Number: 1

  • Number of Pages: XI, 344

  • Number of Illustrations: 1 b/w illustrations

  • Topics: Algebra

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