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Inverses of $r$-primitive $k$-normal elements over finite fields (2201.11334v1)

Published 27 Jan 2022 in math.NT and math.RA

Abstract: Let $r$, $n$ be positive integers, $k$ be a non-negative integer and $q$ be any prime power such that $r\mid qn-1.$ An element $\alpha$ of the finite field $\mathbb{F}{qn}$ is called an {\it $r$-primitive} element, if its multiplicative order is $(qn-1)/r$, 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 $xn-1$ 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 $\mathbb{F}{qn}$, 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 $\mathbb{F}{qn}$ over $\mathbb{F}q$, where $q$ and $n$ are such that $r\mid qn-1$ and there exists a $k$-degree polynomial $g(x)\mid xn-1$ over $\mathbb{F}_q$. In particular, we discuss the existence of an element $\alpha$ in $\mathbb{F}{qn}$ such that $\alpha$ and $\alpha{-1}$ both are simultaneously $1$-primitive and $1$-normal over $\mathbb{F}_q$.

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