A little constructive algebra

EDUARDO J. DUBUC

Colonia del Sacramento, Uruguay.

Congreso; Coloquio Latinoamericano de Algebra; 2005

A LITTLE CONSTRUCTIVE ALGEBRA Mathematics stop being constructive when it incorporate set theory, particularly the axiom of choice and the excluded middle. Van der Waerden wrote in his classic book "Modern Algebra": **I have tried to avoid as much as possible any questionable set-theoretical reasoning in algebra. Unfortunately, a completely finite presentation of algebra, avoiding all non constructive proofs, is not possible without great sacrifices. On the other hand, it was possible at least to compile the building stones for a constructive foundation of algebra. In the theory of fields I did so by presenting the field theoretical operations in a finite number of steps in such a fashion that the intuitionistic foundations of the theory can be seen readily.** The theory of factorization is likewise presented in a more finite manner.'' In this talk we shall illustrate this intuitionistic approach in the context of Gauss's Theorem, developed in an specially constructive manner by Kronecker. We shall also present a construction by A. Joyal of the Pitagorean closure of a totally ordered field without having to decide if a given element has or has not already an square root in the field. A constructive mathematics in the style of the nineteen century is pertinent today because of the use of computers in algebra.