A closer look at some cyclic semifields

We show that different choices of generators $\sigma$ of the Galois group of $\mathbb{F}_{q^n}/\mathbb{F}_{q}$ produce non-isomorphic cyclic semifields $\mathbb{F}_{q^n}[t;\sigma]/\mathbb{F}_{q^n}[t;\sigma](t^m-a)$ when $n\geq m-1$: there are thus $\varphi(n)$ non-isomorphic classes of Sandler semifields $\mathbb{F}_{q^n}[t;\sigma]/\mathbb{F}_{q^n}[t;\sigma](t^m-a)$, one class for each generator $\sigma$ involved in their construction, where $\varphi$ is the Euler function. We prove that when $n=m$, two Sandler semifields constructed from different generators $\sigma_1$ and $\sigma_2$ of ${\rm Gal}(\mathbb{F}_{q^n}/\mathbb{F}_{q})$ are not isotopic. Hence when $n=m$ there are $\varphi(m)$ non-isotopic classes of these semifields, each class belonging to one choice of generator. We then present a full parametrization of the non-isomorphic Sandler semifields $\mathbb{F}_{q^m}[t;\sigma]/\mathbb{F}_{q^m}[t;\sigma](t^m-a)$ , when $m$ is prime and $\mathbb{F}_{q}$ contains a primitive $m$th root of unity. Since for $m=n$, two Sandler semifields constructed from the same generator are isotopic if and only if they are isomorphic, this parametrizes these Sandler semifields up to isotopy, and thus parametrizes both the corresponding non-Desarguesian projective planes, and maximum rank distance codes. Most of our results are proved in all generality for any cyclic Galois field extension.