# Contributions to the theory of Beidleman near-vector spaces

Thesis (PhD)--Stellenbosch University, 2019.

Thesis

ENGLISH SUMMARY: (Please refer to the abstract on the full text for symbols that did not translate well into this abstract). The study of nearfields was started in 1905 by L.E. Dickson. This thesis is a first step toward a detailed study of J.C. Beidleman near-vector spaces, as first introduced by Beidleman in 1966. Recalling well-known results, we conduct a detailed study of finite nearfields by showing how to construct a finite Dickson nearfield and presenting the center of a finite Dickson nearfield that arises from the Dickson pair (q, n). Furthermore, as main results of this thesis, we present the following. We characterise the finite dimensional Beidleman near-vector spaces. We develop an algorithm called EGE (Expanded Gaussian Elimination) which determines the smallest R-subgroup containing a given finite set of vectors v1, . . . , vk 2 Rm where R is a proper nearfield and k,m are positive integers, defined as gen(v1, . . . , vk). We also classify all the subspaces of Rm by designing an algorithm called the Adjustment of the EGE algorithm. We study the concept of seed number of an R-subgroup T (i.e., the minimal cardinality of all the possible finite sets of vectors that generate T) and R-dimension of gen(v1, . . . , vk) (i.e., the number of vectors obtained after the implementation of the EGE algorithm on the finite set of vectors v1, . . . , vk). We evaluate the seed number of Rm for some positive integer m satisfying m jRj +1. Furthermore from the EGE algorithm we also study, for a given pair (a, b) in R2, the generalized distributive set defined as D(a, b) = l 2 R : (a + b) l = a l + b l , where ” ” is the multiplication of the nearfield. We find that in contrast to the situation of D(R) = fl 2 R : (a+ b) l = a l+ b l for all a, b 2 Rg from the work of Zemmer in 1964, the generalized distributive set D(a, b) is not always a subnearfield of R where R is a finite Dickson nearfield arising from the Dickson pair (q, n). We find a sufficient condition on a and b such that D(a, b) is a subfield of the finite field of order qn and develop an algorithm that tests whether D(a, b) is a subfield of Fqn or not. We then investigate D(a, b) where a, b and a + b are all in distinct g qi1 q1 H (where g is a generator of F q n and H is the subgroup generated by gn) and we obtain a construction of a subfield of Fqn by making use of D(a, b).