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

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 ${\mathbb{F}}_{q^n}$.