Primitive Element Pairs with a Prescribed Trace in the Quartic Extension of a Finite Field

Abstract: In this article, we give a largely self-contained proof that the quartic extension $\mathbb{F}{q^4}$ of the finite field $\mathbb{F}_q$ contains a primitive element $\alpha $ such that the element $\alpha+\alpha^{-1}$ is also a primitive element of ${\mathbb{F}{q^4}},$ and $Tr_{\mathbb{F}{q^4}|\mathbb{F}{q}}(\alpha)=a$ for any prescribed $a \in \mathbb{F}_q$. The corresponding result for finite field extensions of degrees exceeding 4 has already been established by Gupta, Sharma and Cohen.