A geometric construction of finite semifields

Abstract. We give a geometric construction of a finite semifield from a certain config-uration of two subspaces with respect to a Desarguesian spread in a finite dimensional vector space over a finite field, and prove that any finite semifield can be obtained in this way. Although no new semifield planes are constructed here, we give explicit sub-spaces from which some known families of semifields can be constructed. In 1965 Knuth [12] showed that each finite semifield generates in total six (not necessarily pairwise non-isotopic) semifields. In certain cases, the geometric construction obtained here allows one to construct another six (not necessarily pairwise non-isotopic) semifields, which may or may not be isotopic to any of the six semifields obtained by Knuth’s operations. Explicit formulas are calculated for the multiplications of the twelve semifields associated with a semifield that is of rank two over its left nucleus. 1