The six semifield planes . . .

In 1965 Knuth [17] noticed that a finite semifield was determined by a 3-cube array (aijk) and that any permutation of the indices would give another semifield. In this article we explain the geometrical significance of these permutations. It is known that a pair of functions (f, g) where f and g are functions from GF (q) to GF (q) with the property that f and g are linear over some subfield and g(x)2+4xf(x) is a non-square for all x ∈ GF (q)∗, q odd, give rise to certain semifields, one of which is commutative of rank 2 over its middle nucleus, one of which arises from a semifield flock of the quadratic cone, and another that comes from a translation ovoid of Q(4, q). We show that there are in fact six non-isotopic semifields that can be constructed from such a pair of functions, which will give rise to six non-isomorphic semifield planes, unless (f, g) are of linear type or of Dickson-Kantor-Knuth type. These six semifields fall into two sets of three semifields related by Knuth operations