Primitive element pairs with one prescribed trace over a finite field

In this article, we establish a sufficient condition for the existence of a primitive element alpha is an element of F-qn such that the element alpha+alpha(-1) is also a primitive element of F-qn, and Tr (Fqn vertical bar Fq) (alpha) = alpha for any prescribed alpha is an element of Fq, where q = p(k) for some prime p and positive integer k. We prove that every finite field F-qn (n >= 5), contains such primitive elements except for finitely many values of q and n. Indeed, by computation, we conclude that there are no actual exceptional pairs (q, n) for n >= 5. (C) 2018 Elsevier Inc. All rights reserved.