Primitive elements with prescribed traces

Abstract

Given a prime power q and a positive integer n, let Fqn denote the finite field with qn elements. Also let a,b be arbitrary members of the ground field Fq. We investigate the existence of a non-zero element ξ∈Fqn such that ξ+ξ−1 is primitive and T(ξ)=a, T(ξ−1)=b, where T(ξ) denotes the trace of ξ in Fq. This was a question intended to be addressed by Cao and Wang (2014). Their work dealt instead with another problem already in the literature. Our solution deals with all values of n≥5.A related study involves the cubic extension Fq3 of Fq. We show that if q≥8⋅1012 then, for any a∈Fq, we can find a primitive element ξ∈Fq3 such that ξ+ξ−1 is also a primitive element of Fq3, and for which the trace of ξ is equal to a. This improves a result of Cohen and Gupta (2021). Along the way we prove a hybridised lower bound on prime divisors in various residue classes, which may be of interest to related existence questions.