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

TL;DR

This paper proves the existence of primitive 1-normal elements in finite fields for all prime powers q and integers n ≥ 3, completing a previously posed problem and extending known results.

Contribution

It establishes the existence of primitive 1-normal elements in all finite fields with n ≥ 3, generalizing prior partial results and solving an open problem.

Findings

Primitive 1-normal elements exist for all q and n ≥ 3.

Completes the proof of a conjecture from previous work.

Extends the class of known primitive elements in finite fields.

Abstract

An element $\alpha \in \mathbb F_{q^n}$ is \emph{normal} if $\mathcal{B} = \{\alpha, \alpha^q, \ldots, \alpha^{q^{n-1}}\}$ 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 $q \geq 3$ 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 $n \geq 3$, thus solving Problem 6.3 from Huczynska, et al (2013).