The zero divisors of the Cayley-Dickson algebras over the real numbers

In this paper we describe algebraically the zero divisors of the Cayley Dickson algebras An = R n for n ≥ 4 over the real numbers. Introduction. The Cayley–Dickson algebra An over R is an algebra structure on R2n given inductively by the formulae: Let x = (x1, x2) and y = (y1, y2) in R n = R2 × R2 then xy = (x1y1 − y2x2, y2x1 + x2y1) where x = (x1,−x2). Therefore A0 = R,A1 = C complex numbers, A2 = H the Hamilton quaternions, A3 = O the Cayley octonians, etc. This four algebras: R,C,H and O are known as the classical Cayley–Dickson algebras and their main distinctive feature of these is: Hurwitz Theorem: Let “‖ ‖” denote the euclidean norm in R2n . Then Classification 1991 AMS 17A99 Accepted for publication at Bol. Soc. Mat. Mex.