Semifields of order q6 with left nucleus Fq3 and center Fq

In [G. Marino, O. Polverino, R. Trombetti, On Fq-linear sets of PG(3,q3) and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families Fi (i=0,…,5) of semifields of order q6 with left nucleus Fq3 and center Fq, according to the different geometric configurations of the associated Fq-linear sets. In this paper we first prove that any semifield of order q6 with left nucleus Fq3, right and middle nuclei Fq2 and center Fq is isotopic to a cyclic semifield. Then, we focus on the family F4 by proving that it can be partitioned into three further non-isotopic families: , , and we show that any semifield of order q6 with left nucleus Fq3, right and middle nuclei Fq2 and center Fq belongs to the family .