A note on generalized distributive sets of a finite Dickson nearfield

RESUMO

Let $\mathbb{F}_{q^n}$ be a finite field where $q=p^l$ ($p$ is a prime) and a Dickson nearfield $R$ associated to the field. In this note we investigate the basic properties of the generalized distributive set $D(a,b)$ where $a,b\in R$. The set is well-known when $n=1, 2$. For $n>2$, we characterize when $D(a,b)$ is a subfield of $\mathbb{F}_{q^n}$. Furthermore we find the unique largest subfield of $\mathbb{F}_{q^n}$ contained in $D(a,b)$. We also provide a condition to guarantee that $D(a,b)$ is not a subfield of $\mathbb{F}_{q^n}$. We end the note by posing the question of when $D(a,b)$, as a space over $\mathbb{F}_p$, has a primitive normal basis.