Van Der Waerden's Construction of a Splitting Field

Abstract

In his classic book, Modern Algebra, Van der Waerden (1953) gave a procedure for factoring polynomials over a finite-dimensional, separable, simple extension field. I believe that there is a nonconstructive component to his proof, and I will indicate where it comes in and why. Although I'm sure that this component could be avoided while staying within the framework that he set down, it is simpler to get around the problem by working with a splitting algebra, which is easily constructed for any polynomial and base field. The existence of a splitting field then follows from Van der Waerden's argument.

Keywords:

• Constructive mathematics
• Splitting filed

Mathematics Subject Classification:

• Primary 12F05
• Secondary 03F65

Notes

Communicated by I. Swanson.