Existence of primitive $1$-normal elements in finite fields

Abstract

An element $\alpha \in {\mathbb{F}}_{q^n}$ is \emph{normal} if $\mathcal{B}=\{\alpha ,{\alpha }^q,\dots ,{\alpha }^{q^{n-1}}\}$ forms a basis of ${\mathbb{F}}_{q^n}$ as a vector space over ${\mathbb{F}}_q$; in this case, $\mathcal{B}$ is a normal basis of ${\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q$. The notion of $k$-normal elements was introduced in Huczynska et al (2013). Using the same notation as before, $\alpha$ is $k$-normal if $\mathcal{B}$ spans a co-dimension $k$ subspace of ${\mathbb{F}}_{q^n}$. It can be shown that $1$-normal elements always exist in ${\mathbb{F}}_{q^n}$, and Huczynska et al (2013) show that elements that are simultaneously primitive and $1$-normal exist for $q\ge 3$ and for large enough $n$ when $\gcd (n,q)=1$ (we note that primitive $1$-normals cannot exist when $n=2$). In this paper, we complete this theorem and show that primitive, $1$-normal elements of ${\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q$ exist for all prime powers $q$ and all integers $n\ge 3$, thus solving Problem 6.3 from Huczynska, et al (2013).

29 pages