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 (\log q^n) (\log\log q^n)^{ 1+\varepsilon}) $, where $\varepsilon>0$ is a small number, contains a primitive normal element.
Categories: math.GM