The translate and line properties for 2-primitive elements in quadratic extensions

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 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 $(q^n-1)/r$. Likewise, if, in the above definition, $\alpha$ is restricted to the value $1$, we say that ${\mathbb{F}}_{q^n}/{\mathbb{F}}_q$ possesses the translate property. In this paper we take $r=n=2$ (so that necessarily $q$ is odd) and prove that ${\mathbb{F}}_{q^2}/{\mathbb{F}}_q$ possesses the translate property for 2-primitive elements unless $q\in \{5,7,11,13,31,41\}$. With some additional theoretical and computational effort, we show also that ${\mathbb{F}}_{q^2}/{\mathbb{F}}_q$ possesses the line property for 2-primitive elements unless $q\in \{3,5,7,9,11,13,31,41\}$.

arXiv admin note: text overlap with arXiv:1906.08046, arXiv:1903.03160