Alternative algebras over an arbitrary field

Abstract

The results of M. Zorn concerning alternative algebras 2 are incomplete over modular fields since, in his study of alternative division algebras, Zorn restricted the characteristic of the base field to be not two or three. In this paper we present first a unified treatment of alternative division algebras which, together with Zorn's results, permits us to state that any alternative, but not associative, algebra A over an arbitrary field F is central simple (that is, simple for all scalar extensions) if and only if A is a Cayley-Dickson algebra 3 over F.