About $r$- primitive and $k$-normal elements in finite fields

TL;DR

This paper investigates the existence of elements in finite fields that are both r-primitive and k-normal, generalizing normal elements, and provides theoretical results along with a numerical example in characteristic 11.

Contribution

It introduces the concept of r-primitive, k-normal elements in finite fields and establishes general existence results, extending the theory of normal elements.

Findings

Existence results for r-primitive, k-normal elements in finite fields.

General theorems applicable to various parameters r, k, q, n.

Numerical example demonstrating the theory in characteristic 11.

Abstract

In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of $k$-normal elements: an element $\alpha \in \mathbb{F}_{q^n}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g_{\alpha}(x)= \alpha x^{n-1}+\alpha^qx^{n-2}+\ldots +\alpha^{q^{n-2}}x+\alpha^{q^{n-1}}$ and $x^n-1$ in $\mathbb{F}_{q^n}[x]$ has degree $k$, generalizing the concept of normal elements (normal in the usual sense is $0$-normal). In this paper we discuss the existence of $r$-primitive, $k$-normal elements in $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$, where an element $\alpha \in \mathbb{F}_{q^n}^*$ is $r$-primitive if its multiplicative order is $\frac{q^n-1}{r}$. We provide many general results about the existence of this class of elements and we work a numerical example over finite fields of characteristic $11$.