Alternative Algebras over an Arbitrary Field

R. D. SCHAFER The results of M. Zorn concerning alternative algebras 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 over F. A. A. Albert in a recent paper, Non-associative algebras I : Fundamental concepts and isotopy,* introduced the concept of isotopy for the study of non-associative algebras. We present in the concluding section of this paper theorems concerning isotopes (with unity quantities) of alternative algebras. The reader is referred to Albert's paper, moreover, for definitions and explanations of notations which appear there and which, in the interests of brevity, have been omitted from this paper.