Existence of primitive k-normal elements for critical values over finite fields

TL;DR

This paper investigates the existence of primitive k-normal elements in finite fields specifically at the critical value k = n/2, extending previous work and providing a complete characterization for certain cases.

Contribution

It establishes new general results on primitive k-normal elements at the critical value k = n/2 and characterizes primitive 3-normal elements in _{q^6} over _q.

Findings

Proved existence results for primitive k-normal elements at k = n/2.

Provided a complete characterization of primitive 3-normal elements in _{q^6}.

Extended understanding of primitive k-normal elements beyond the previously studied cases.

Abstract

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements. An element $\alpha \in \mathbb{F}_{q^n}$ is called $k$-normal over $\mathbb{F}_q$ if $\alpha$ and its conjugates generate a vector subspace of $\mathbb{F}_{q^n}$ of dimension $n-k$ over $\mathbb{F}_q$. The existence of primitive $k$-normal elements and related properties have been studied throughout the past few years for $k > n/2$. In this paper, we provide general results on the existence of primitive $k$-normal elements for the critical value $k = n/2$, which have not been studied until now, except for $n = 4$. Furthermore, we show the strength of this result by providing a complete characterization of the existence of primitive $3$-normal elements in $\mathbb{F}_{q^6}$ over $\mathbb{F}_q$.