Primitive elements with prescribed trace

Let $$q$$q be a power of a prime number $$p$$p. Let $$n$$n be a positive integer. Let $$\mathbb {F}_{q^n}$$Fqn denote a finite field with $$q^n$$qn elements. In this paper, we consider the existence of the some specific elements in the finite field $$\mathbb {F}_{q^n}$$Fqn. We get that when $$n\ge 29$$n≥29, there are elements $$\xi \in \mathbb {F}_{q^n}$$ξ∈Fqn such that $$\xi +\xi ^{-1}$$ξ+ξ-1 is a primitive element of $$\mathbb {F}_{q^n}$$Fqn, and $$\mathrm{Tr}(\xi ) = a, \mathrm{Tr}(\xi ^{-1}) = b$$Tr(ξ)=a,Tr(ξ-1)=b for any pair of prescribed $$a, b \in \mathbb {F}_q^*$$a,b∈Fq∗.