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

Abstract

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 primitive element $\hat{I}¾$ such that $\hat{I}¾+\hat{I}{¾}^{-1}$ is also primitive, and ${\text{Tr}}_{{\mathbb{F}}_{q^3}/{\mathbb{F}}_q}(\hat{I}¾)=a$ 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 arbitrary degree $n\ge 3$.

4 pages