Some Applications of Galois Theory to Normal Polynomials

Abstract

In his classical text Algebra [8, p. 124], van der Waerden introduces the concept of a normal polynomial f(X) over a field Kan irreducible polynomial over K such that K(O) is a splitting field of f(X) over K for each root 0 of f(X) in an extension field of K. But like other authors (see, for example, [3, p. 259]) van der Waerden does little with the concept, and indeed, at the point at which the definition of a normal polynomial is introduced in [8], the student has little to work with no Galois theory and only the rudiments of field theory. On the other hand, once the Fundamental Theorem of Galois Theory has been proved and cyclotomic fields have been studied in Section 8.4 of [8], some nice applications of these results to the theory of normal polynomials can be given. The purpose of this note is to present some of these applications. We make no claims for originality of the results, although we have not been able to find some of them in the literature. Also, to be a bit more concrete, we restrict to the case where the base field is the field Q of rational numbers. Our main purpose is to outline proofs of the following two theorems.