A generalisation of Dickson's commutative division algebras

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.