Jim wrote:

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Rich wrote:

Is there an interesting cubic extension of the octonians?

I'm not sure what Rich means by a "cubic" extension, but there's an extension of the octonians called the sedenions, and some folks go even farther.

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Please, folks: it's spelled "octonions". (I care only because they're sacred!) Nor do I know what Rich means by "a cubic extension". (The abstract Jim posted was, um, rather abstract.) But indeed there is a doubling procedure ("Cayley-Dickson") that, repeatedly applied, starting with the reals, gives the complexes, then the quaternions, then the octonions, and can be continued indefinitely to give a real algebra of dimension 2^n, for each n. (A *real algebra* of dimension k is just a bilinear function R^k x R^k -> R.) (Cf. http://planetmath.org/encyclopedia/DoublingProcess.html. Note they use "non-associative algebra" to mean a "real algebra" as above; it isn't *required* to be associative, but is allowed to be.) But for n >= 4, all these algebras have zero-divisors. And a theorem of J. Frank Adams, c. 1956 -- that AFAIK has only a topological proof and no known purely algebraic one -- shows that real algebras without zero-divisors (aka real division algebras) exist *only* in dimensions 1,2,4,8. (Surprisingly, there are real division algebras in dimensions 2,4,8 other than the complexes, quaternions, or octonions.)