Locally finite near-fields

Abstract

A near-field is locally finite if every finite subset of it generates a finite sub-near-field. The main aim of this thesis is to give a coherent account of locally finite-near-fields, including finite ones. The well known results for finite near-fields are lised and proofs are given where appropriate. The results of Zassenhaus classify finite regular near-fields according to their order, pln, and the order of centres, pl, and Luneburg has determined the number of isomorphism types within each class. A polynomial h is given here which, together with the triple p, l, n, completely determines a finite regular near-field, up to isomorphism. The sub-near-field structure is determined in terms of these invariants and some results concerning near-field embeddings are included.