  Cayley-Dickson algebras are an infinite sequence of non-associative algebras
starting with the reals, complexes, quaternions, and octonions. We study the
zero-divisors in the higher Cayley-Dickson algebras. In particular, we show
that the annihilator of any element in the 2^n-dimensional Cayley-Dickson
algebra has dimension at most 2^n-4n+4. Moroever, every multiple of four
between 0 and this upper bound actually occurs as the dimension of some
annihilator (a theorem of Moreno says that only multiples of four can occur).
We completely describe all the zero-divisors whose annihilator has dimension
2^n-4n+4.
