Primitive Element Pairs with One Prescribed Trace over a Finite Field

Abstract

In this article, we establish a sufficient condition for the existence of a primitive 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 $T{r}_{{\mathbb{F}}_{q^n}|{\mathbb{F}}_q}(\alpha )=a$ for any prescribed $a\in {\mathbb{F}}_q$, where $q=p^k$ for some prime $p$ and positive integer $k$. We prove that every finite field ${\mathbb{F}}_{q^n}(n\ge 5),$ contains such primitive elements except for finitely many values of $q$ and $n$. Indeed, by computation, we conclude that there are no actual exceptional pairs $(q,n)$ for $n\ge 5.$

19 pages, 1 table