Further results on the Morgan-Mullen conjecture

Abstract

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 ${\mathbb{F}}_{q^n}/{\mathbb{F}}_q$ 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 $q\le n\le O(q^{\epsilon })$, where $\epsilon =2$ for the asymptotic results and $\epsilon =1.25$ for the effective ones. For $n$ even we need to assume that $q-1\nmid n$.

arXiv admin note: text overlap with arXiv:1709.03141