Inverses of $r$-primitive $k$-normal elements over finite fields

TL;DR

This paper investigates the existence of elements in finite fields that are both $r$-primitive and $k$-normal, and whose inverses share these properties, providing conditions and demonstrating their existence in most cases.

Contribution

It introduces a characteristic function for $k$-normal elements and establishes new sufficient conditions for the simultaneous existence of such elements and their inverses in finite fields.

Findings

Existence of elements with both properties under certain conditions

Almost always, such elements exist for large enough fields

Conditions involving divisibility and polynomial factors are sufficient

Abstract

Let $r$, $n$ be positive integers, $k$ be a non-negative integer and $q$ be any prime power such that $r\mid q^n-1.$ An element $\alpha$ of the finite field $\mathbb{F}_{q^n}$ is called an {\it $r$-primitive} element, if its multiplicative order is $(q^n-1)/r$, and it is called a {\it $k$-normal} element over $\mathbb{F}_q$, if the greatest common divisor of the polynomials $m_\alpha(x)=\sum_{i=1}^{n} \alpha^{q^{i-1}}x^{n-i}$ and $x^n-1$ is of degree $k.$ In this article, we define the characteristic function for the set of $k$-normal elements, and with the help of this, we establish a sufficient condition for the existence of an element $\alpha$ in $\mathbb{F}_{q^n}$, such that $\alpha$ and $\alpha^{-1}$ both are simultaneously $r$-primitive and $k$-normal over $\mathbb{F}_q$. Moreover, for $n>6k$, we show that there always exists an $r$-primitive and $k$-normal element $\alpha$ such that $\alpha^{-1}$ is also $r$-primitive and $k$-normal in all but finitely many fields $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$, where $q$ and $n$ are such that $r\mid q^n-1$ and there exists a $k$-degree polynomial $g(x)\mid x^n-1$ over $\mathbb{F}_q$. In particular, we discuss the existence of an element $\alpha$ in $\mathbb{F}_{q^n}$ such that $\alpha$ and $\alpha^{-1}$ both are simultaneously $1$-primitive and $1$-normal over $\mathbb{F}_q$.