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We prove that for any prime power $q\notin\{3,4,5\}$, the cubic extension$\mathbb{F}_{q^3}$ of the finite field $\mathbb{F}_q$ contains a primitiveelement $\xi$ such that $\xi+\xi^{-1}$ is also primitive, and$\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(\xi)=a$ for any prescribed$a\in\mathbb{F}_q$. This completes the proof of a conjecture of Gupta, Sharma,and Cohen concerning the analogous problem over an extension of arbitrarydegree $n\ge3$.

Primitive element pairs with a prescribed trace in the cubic extension of a finite field