When Does F(α d ) = F(α)?

Abstract

Suppose that α is algebraic over a field . A standard exercise in a first course in field theory is to show that if the degree of the minimal polynomial of α over is odd, then . In this note, we generalize the sufficient conditions on the minimal polynomial for α so that for any particular integer . Then, given any finite set of integers , this generalization allows us to construct irreducible polynomials f, with , such that for all .