Some applications of nearfields

Abstract

Nearfields and near-rings are related to many other structures and needed for several representation theorems. Therefore it is important to gain knowledge about the structure of near-rings and nearfields and to find construction methods. The first examples of proper nearfields were constructed by L.E. Dickson 1905, they were finite. 30 years later H. Zassenhaus completely determined the finite nearfields his attention having been attracted to them by the study of certain permutation groups. By the axiomatization of Dickson's methods, H. Karzel succeeded in giving new examples of infinite nearfields. In the extensive generalization by J. Timm the so-called “Dicksonian processes” are the most important tool for constructions of nearfields and near-rings. The report (44) by H. Wähling is a summary of the results on nearfields obtained so far.