# On the generalized distributive set of a finite nearfield

## TL;DR

This paper investigates the structure of generalized distributive sets in finite Dickson nearfields, revealing they are not generally subfields or subnearfields, and provides conditions and algorithms for their identification, along with subgroup dimension analysis.

## Contribution

It introduces the concept of generalized distributive sets in finite Dickson nearfields, analyzes their algebraic structure, and develops algorithms to determine when these sets form subfields.

## Key findings

- D(α, β) is not generally a subfield of the finite field.
- D(α, β) is not generally a subnearfield of R.
- Conditions are provided for D(α, β) to be a subfield.

## 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 $v_2$.