Semifields of order $q^6$ with left nucleus ${\mathbb{F}}_{q^3}$ and center ${\mathbb{F}}_q$

Abstract

In [G. Marino, O. Polverino, R. Trombetti, On ${\mathbb{F}}_q$-linear sets of $\mathit{PG}(3,q^3)$ and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families ${\mathcal{F}}_i$ ($i=0,\dots ,5$) of semifields of order $q^6$ with left nucleus ${\mathbb{F}}_{q^3}$ and center ${\mathbb{F}}_q$, according to the different geometric configurations of the associated ${\mathbb{F}}_q$-linear sets. In this paper we first prove that any semifield of order $q^6$ with left nucleus ${\mathbb{F}}_{q^3}$, right and middle nuclei ${\mathbb{F}}_{q^2}$ and center ${\mathbb{F}}_q$ is isotopic to a cyclic semifield. Then, we focus on the family ${\mathcal{F}}_4$ by proving that it can be partitioned into three further non-isotopic families: ${\mathcal{F}}_4^{(\mathrm{a})}$, ${\mathcal{F}}_4^{(\mathrm{b})}$, ${\mathcal{F}}_4^{(\mathrm{c})}$ and we show that any semifield of order $q^6$ with left nucleus ${\mathbb{F}}_{q^3}$, right and middle nuclei ${\mathbb{F}}_{q^2}$ and center ${\mathbb{F}}_q$ belongs to the family ${\mathcal{F}}_4^{(\mathrm{c})}$.