Existence of primitive $2$-normal elements in finite fields

TL;DR

This paper proves the existence of primitive 2-normal elements in finite fields for all cases, extending the theory of normal elements and providing new conditions using Gauss sum estimates and computational methods.

Contribution

It completely solves the existence problem for primitive 2-normal elements in finite fields, a case previously unresolved, and introduces new criteria for k-normal element existence.

Findings

Proves existence of primitive 2-normal elements for all finite fields.

Develops new conditions for the existence of k-normal elements.

Uses Gauss sum estimates and computational techniques to achieve results.

Abstract

An element $\alpha \in \mathbb{F}_{q^n}$ is normal over $\mathbb{F}_q$ if $\mathcal{B}=\{\alpha, \alpha^q, \alpha^{q^2}, \cdots, \alpha^{q^{n-1}}\}$ forms a basis of $\mathbb{F}_{q^n}$ as a vector space over $\mathbb{F}_q$. It is well known that $\alpha \in \mathbb{F}_{q^n}$ is normal over $\mathbb{F}_q$ if and only if $g_{\alpha}(x)=\alpha x^{n-1}+\alpha^q x^{n-2}+ \cdots + \alpha^{q^{n-2}}x+\alpha^{q^{n-1}}$ and $x^n-1$ are relatively prime over $\mathbb{F}_{q^n}$, that is, the degree of their greatest common divisor in $\mathbb{F}_{q^n}[x]$ is $0$. Using this equivalence, the notion of $k$-normal elements was introduced in Huczynska et al. ($2013$): an element $\alpha \in \mathbb{F}_{q^n}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g_{\alpha}[x]$ and $x^n-1$ in $\mathbb{F}_{q^n}[x]$ has degree $k$; so an element which is normal in the usual sense is $0$-normal. Huczynska et al. made the question about the pairs $(n,k)$ for which there exist primitive $k$-normal elements in $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$ and they got a partial result for the case $k=1$, and later Reis and Thomson ($2018$) completed this case. The Primitive Normal Basis Theorem solves the case $k=0$. In this paper, we solve completely the case $k=2$ using estimates for Gauss sum and the use of the computer, we also obtain a new condition for the existence of $k$-normal elements in $\mathbb{F}_{q^n}$.