Semifields from skew polynomial rings

Abstract

Abstract Skew polynomial rings are used to construct finite semifields, following from a construction of Ore and Jacobson of associative division algebras. Johnson and Jha [10] 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 [13] and implicitly by Dempwolff in [2].