On the generalized distributive set of a finite nearfield

Abstract

For any nearfield $(R,+,\circ )$, denote by $D(R)$ the set of all distributive elements of $R$. Let $R$ be a finite Dickson nearfield that arises from Dickson pair $(q,n)$. For a given pair $(\alpha ,\beta )\in R^2$ we study the generalized distributive set $D(\alpha ,\beta )$ where $"\circ "$ is the multiplication of the Dickson nearfield. We find that $D(\alpha ,\beta )$ is not in general a subfield of the finite field ${\mathbb{F}}_{q^n}$. In contrast to the situation for $D(R)$, we also find that $D(\alpha ,\beta )$ is not in general a subnearfield of $R$. We obtain sufficient conditions on $\alpha ,\beta$ for $D(\alpha ,\beta )$ to be a subfield of ${\mathbb{F}}_{q^n}$ and derive an algorithm that tests if $D(\alpha ,\beta )$ is a subfield of ${\mathbb{F}}_{q^n}$ or not. We also study the notions of $R$-dimension, $R$-basis, seed sets and seed number of $R$-subgroups of the Beidleman near-vector spaces $R^m$ where $m$ is a positive integer. Finally we determine the maximal $R$-dimension of $gen(v_1,v_2)$ for $v_1,v_2\in R^m$, where $gen(v_1,v_2)$ is the smallest $R$-subgroup containing the vectors $v_1$ and $v_2$.