On the Hughes-Kleinfeld and Knuth's semifields two-dimensional over a weak nucleus

In 1960 Hughes and Kleinfeld [4] constructed a finite semifield which is two dimensional over a weak nucleus. In 1965 Knuth [7] constructed a further three families of finite semifields which are also two dimensional over a weak nucleus. Moreover, in the same article, Knuth describes operations that allow one to obtain up to six semifields from a given semifield. We show how these operations in fact relate these four families of finite semifields and that up to isotopy there are two families, one which generates at most two non-isotopic semifields under the Knuth operations and the other which generates at most three non-isotopic semifields.

• Type: Conference
• Year: 2007
• Where: DCC
• Authors: Simeon Ball, Michel Lavrauw
• Keywords: Computer Graphics, DCC 2007, Knuth Describes Operations, Weak Nucleus