Semifields from skew polynomial rings

TL;DR

This paper explores the relationship between two constructions of finite semifields using skew polynomial rings, demonstrating their isotopy, simplifying nucleus calculations, and providing improved bounds on isotopism classes.

Contribution

It shows that Petit’s and Jha-Johnson’s semifield constructions are isotopic, introduces a method to compute nuclei more easily, and improves bounds on the number of isotopism classes.

Findings

Petit and Jha-Johnson semifields are isotopic.

Skew polynomial rings facilitate nucleus calculations.

New upper bounds for isotopism classes are established.

Abstract

Skew polynomial rings were used to construct finite semifields by Petit in 1966, following from a construction of Ore and Jacobson of associative division algebras. In 1989 Jha and Johnson constructed the so-called cyclic semifields, obtained using irreducible semilinear transformations. In this work we show that these two constructions in fact lead to isotopic semifields, show how the skew polynomial construction can be used to calculate the nuclei more easily, and provide an upper bound for the number of isotopism classes, improving the bounds obtained by Kantor and Liebler in 2008 and implicitly in recent work by Dempwolff.