Primitive elements with prescribed traces

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