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 $T{r}_{{\mathbb{F}}_{q^4}|{\mathbb{F}}_q}(\alpha )=a$ 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.

14 pages