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

Abstract

In 1960 Hughes and Kleinfeld (Am J Math 82:389---392, 1960) constructed a finite semifield which is two-dimensional over a weak nucleus, given an automorphism ? of a finite field $$\mathbb {K}$$ and elements $$\mu, \eta \in \mathbb{K}$$ with the property that $$x^{\sigma+1}+\mu{x} - \eta$$ has no roots in $$\mathbb{K}$$ . In 1965 Knuth (J Algebra 2:182---217, 1965) constructed a further three finite semifields which are also two-dimensional over a weak nucleus, given the same parameter set $$(\mathbb{K},\sigma,\mu,\eta)$$ . 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 finite semifields, for a fixed parameter set, and yield at most five non-isotopic semifields out of a possible 24. These five semifields form two sets of semifields, one of which consists of at most two non-isotopic semifields related by Knuth operations and the other of which consists of at most three non-isotopic semifields.