A closer look at some cyclic semifields

TL;DR

This paper classifies and parametrizes non-isomorphic cyclic semifields constructed from Galois groups, revealing their structure, isotopy classes, and applications to projective planes and coding theory.

Contribution

It provides a complete classification and parametrization of Sandler semifields based on Galois group generators, especially for prime degrees and specific field conditions.

Findings

Number of non-isomorphic classes equals Euler's totient function of n.

Different Galois generators produce non-isomorphic semifields.

Complete parametrization of semifields when m is prime and field contains a primitive mth root of unity.

Abstract

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.