Sufficient condition for existence of special type of primitive normal elements over finite fields

Abstract: Let $\mathbb{F}{q^n}$ be the extension of the field $\mathbb{F}_q$ of degree n, where $q$ is power of prime $p$, i.e $q=p^k$, where k is a positive integer. In this paper, we provide sufficient condition for the existence of a primitive normal element $\alpha\in\mathbb{F}{q^n} $ such that $\alpha^2+\alpha+1$ is also primitive normal element over $\mathbb{F}_{q^n}$.