Among those algebras whose
multiplication does not satisfy the associative law is a particular family of
noncommutative Jordan algebras, the generalized Cayley-Dickson algebras. These are
certain central simple algebras whose dimensions are all powers of two. Most of this
paper is concerned with giving the classification up to isomorphism of those
of dimensions 16, 32, and 64 and determining the automorphism groups.
In addition to this some generalized Cayley-Dickson division algebras are
constructed. Precise criteria for when the 16-dimensional algebras are division
algebras are formulated and applied to algebras over some common fields. For
higher dimensions no such criteria are given. However, specific examples of
division algebras for each dimension 2t are constructed over power-series
fields.
