A geometric construction of finite semifields

We give a geometric construction of a finite semifield from a certain configuration 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 subspaces from which some known families of semifields can be constructed. In 1965 Knuth [D.E. Knuth, Finite semifields and projective planes, J. Algebra 2 (1965) 182–217] 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 Knuthʹs 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.