Finite field extensions with the line or translate property for $r$-primitive elements

Abstract

Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\mid q^n-1$. We say that the extension ${\mathbb{F}}_{q^n}/{\mathbb{F}}_q$ possesses the line property for $r$-primitive elements property if, for every $\alpha ,\theta \in {\mathbb{F}}_{q^n}^{\ast }$, such that ${\mathbb{F}}_{q^n}={\mathbb{F}}_q(\theta )$, there exists some $x\in {\mathbb{F}}_q$, such that $\alpha (\theta +x)$ has multiplicative order $(q^n-1)/r$. We prove that, for sufficiently large prime powers $q$, ${\mathbb{F}}_{q^n}/{\mathbb{F}}_q$ possesses the line property for $r$-primitive elements. We also discuss the (weaker) translate property for extensions.

arXiv admin note: text overlap with arXiv:1903.03160