On Special Primitive Elements over Finite Fields

Abstract

Let $\mathbb{F}_{q^n}$ be an extension of the field $\mathbb{F}_q$ of degree $n,$ where $q=p^k$ for some prime $p$ and positive integer $k$. %is the characteristic of the field $\mathbb{F}_q.$ In this article, we establish a sufficient condition for the existence of a primitive element $\alpha \in {\mathbb{F}_{q^n}}$ such that $\alpha^2+\alpha +1$ is also primitive and $Tr_{\mathbb{F}_{q^n}|\mathbb{F}_{q}}(\alpha)=a$ for any $a\in \mathbb{F}_q$.