A Course in Constructive Algebra

1. Front Matter

   Pages i-xi

   PDF

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

    PDF

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.

• Galois theory
• algebra
• field
• matrices
• matrix
• torsion
• transformation

• 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

Policies and ethics