On the theorem of Gleason and Marsh
Neal Zierler · Proceedings of the American Mathematical Society · 1958
The order of a nonzero element a of a finite field is the smallest of the positive integers j for which ai = 1. If f(x) is irreducible over K, then all of its roots are of the same order, for given any two roots of f lying in a finite extension F of K there is always an automorphism of F mapping one on the other. We may therefore define the order of the irreducible polynomial f to be the order of any one of its roots. The purpose of this note is to establish the following generalization of the theorem of Gleason and Marsh.3