  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 $q>m$.
