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In this article, we establish a sufficient condition for the existence of aprimitive element $\alpha \in {\mathbb{F}_{q^n}}$ such that the element$\alpha+\alpha^{-1}$ is also a primitive element of ${\mathbb{F}_{q^n}},$ and$Tr_{\mathbb{F}_{q^n}|\mathbb{F}_{q}}(\alpha)=a$ for any prescribed $a \in\mathbb{F}_q$, where $q=p^k$ for some prime $p$ and positive integer $k$. Weprove that every finite field $\mathbb{F}_{q^n}~ (n \geq5),$ contains suchprimitive elements except for finitely many values of $q$ and $n$. Indeed, bycomputation, we conclude that there are no actual exceptional pairs $(q,n)$ for$n\geq5.$

Primitive Element Pairs with One Prescribed Trace over a Finite Field