On the generalized distributive set of a finite nearfield

Abstract

For any nearfield (R, +, •), 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 (α, β) ∈ R 2 we study the generalized distributive set•" is the multiplication of the Dickson nearfield. We find that D(α, β) is not in general a subfield of the finite field F q n . In contrast to the situation for D(R), we also find that D(α, β) is not in general a subnearfield of R. We obtain sufficient conditions on α, β for D(α, β) to be a subfield of F q n and derive an algorithm that tests if D(α, β) is a subfield of 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 ∈ R m , where gen(v 1 , v 2 ) is the smallest R-subgroup containing the vectors v 1 and v 2 .

On the center of a finite Dickson nearfield

“…Zassenhauss [10], Dancs [3,4], Karzel and Ellers [2] have solved some important problems in this area. Recently the author in [5] has investigated on the generalized distributive set of a finite nearfield In the paper [2], Eller and Karzel showed that the center of a finite Dickson that arises from Dickson pair (q, n) is equal to a finite field of order q n . In the present work, we provide a simple and shortest proof of a result due to Eller and Karzel on the presentation of the center of a finite Dickson nearfield that arises from a Dickson pair (q, n).…”