On the existence of primitive completely normal bases of finite fields

Abstract

Let ${\mathbb{F}}_q$ be the finite field of characteristic $p$ with $q$ elements and ${\mathbb{F}}_{q^n}$ its extension of degree $n$. We prove that there exists a primitive element of ${\mathbb{F}}_{q^n}$ that produces a completely normal basis of ${\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q$, provided that $n=p^{\ell }m$ with $(m,p)=1$ and $q>m$.