A generalisation of Dickson's commutative division algebras

TL;DR

This paper generalizes Dickson's construction of commutative division algebras by allowing more flexible algebraic structures, including non-commutative and non-associative cases, and characterizes when these are division algebras.

Contribution

It introduces a broader construction method for division algebras, extending classical Dickson algebras to more general settings with new conditions for division properties.

Findings

Established conditions for when the generalized algebras are division algebras.

Classified when the resulting algebras are non-isomorphic.

Computed automorphism groups for specific cases.

Abstract

Dickson's commutative semifields are an important class of finite division algebras. We generalise Dickson's construction of commutative division algebras by doubling both finite field extensions and central simple algebras and not restricting us to the classical setup where a cyclic field extension is taken. The latter case yields algebras which are no longer commutative nor associative. Conditions for when the algebras are division algebras are given that canonically generalise the classical ones known up to now. We investigate when we obtain non-isomorphic algebras and compute all the automorphisms, including the structure of the automorphism group in some cases.