Small Primitive Normal Elements in Finite Fields

Let $q=p^k$ be a prime power, let $\mathbb{F}_q$ be a finite field and let $n\geq2$ be an integer. This note investigates the existence small primitive normal elements in finite field extensions $\mathbb{F}_{q^n}$. It is shown that a small nonstructured subset $\mathcal{A}\subset \mathbb{F}_{q^n}$ of cardinality $\#\mathcal{A}\gg q^{\varepsilon}$, where $\varepsilon>0$ is a small number, contains a primitive normal element.