Commutative semifields, two dimensional over their middle nuclei

Abstract

Then we say that S is a (finite) semzj?eld. The subset N, = (n E S: (xn)y = x(ny), Vx, y E S} is known as the middle nucleus of S. N, is a field, and S may be regarded as a (left or right) vector space over N,. A semifield in which multiplication is not associative (i.e., N, # S) is called proper. A finite semifield necessarily has prime power order and proper finite semifields exist for all orders q = p’, p prime, where Y > 3 if p is odd and r>4ifp=2. There is a considerable link between finite semifields and finite projective planes of Lenz-Barlotti class V.1. Indeed, any finite proper semilield gives rise to such a plane, and conversely any plane of the above type yields many isotopic, but not necessarily isomorphic, semifields. The paper by Knuth [4] is an excellent survey of finite semifields and their connections with projective planes. We mention one of the examples given in 141‘