A closer look at some cyclic semifields
Susanne Pumpluen
TL;DR
The work addresses how the choice of generator $\sigma$ of a cyclic Galois extension $K/F$ influences the structure of nonassociative cyclic algebras $(K/F,\sigma,a)$ and Sandler's semifields, proving non-isomorphism for distinct generators when $n\ge m-1$ and non-isotopy for $n=m$, with counts given by $\varphi(n)$ and $\varphi(m)$. It provides explicit parametrizations for prime degree $m$ under the condition that the base field contains a primitive $m$-th root of unity, yielding one representative per isomorphism class and connecting these algebras to non-Desarguesian planes and maximum rank distance (MRD) codes. The paper also develops the MRD-code perspective: isotopy classes of semifields correspond to equivalence classes of $\mathbb{F}_q$-linear MRD-codes in $M_{m^2}(\mathbb{F}_q)$ (and in $M_m(\mathbb{F}_{q^m})$), and shows that different choices of generator produce non-equivalent codes. Finally, it extends the parametrization to skew $\sigma$-constacyclic codes via left ideals in Petit algebras, highlighting how code equivalence May depend on the underlying generator choice. The results unify nonassociative algebra, semifield isotopy, projective planes, and MRD-code theory, with concrete parametrizations in odd prime degree and explicit code constructions.
Abstract
We show that different choices of generators $σ$ of the Galois group of $\mathbb{F}_{q^n}/\mathbb{F}_{q}$ produce non-isomorphic cyclic semifields $\mathbb{F}_{q^n}[t;σ]/\mathbb{F}_{q^n}[t;σ](t^m-a)$ when $n\geq m-1$: there are thus $\varphi(n)$ non-isomorphic classes of Sandler semifields $\mathbb{F}_{q^n}[t;σ]/\mathbb{F}_{q^n}[t;σ](t^m-a)$, one class for each generator $σ$ involved in their construction, where $\varphi$ is the Euler function. We prove that when $n=m$, two Sandler semifields constructed from different generators $σ_1$ and $σ_2$ 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 $\mathbb{F}_{q^m}[t;σ]/\mathbb{F}_{q^m}[t;σ](t^m-a)$ , 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.