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
.
Acknowledgment
The author thanks the anonymous referees for their valuable suggestions.