Further results on the Morgan-Mullen conjecture

Let ${\mathbb{F}}_q$ be the finite field of characteristic $p$ with $q$ elements and ${\mathbb{F}}_{q^n}$ its extension of degree $n$. The conjecture of Morgan and Mullen asserts the existence of primitive and completely normal elements (PCN elements) for the extension ${\mathbb{F}}_{q^n}\mathrm{/}{\mathbb{F}}_q$ for any $q$ and $n$. It is known that the conjecture holds for $n\le q$. In this work we prove the conjecture for a larger range of exponents. In particular, we give sharper bounds for the number of completely normal elements and use them to prove asymptotic and effective existence results for $q\leq n\leq O(q^\epsilon)$,where$\epsilon=2$fortheasymptoticresultsand$\epsilon=1.25$fortheeffectiveones\mathrm{.}For$n$evenweneedtoassumethat$q-1\nmid n$.