Re: Why is there no *really* useful Algebra beyond complex numbers?

Actually the reason there are no finite dimensional division algebras
over the real numbers except for dimensions 1, 2, 4, and 8, is a deep
theorem with intimate connections with topology. The quaternions
(dimentsion 4) and Cayley numbers (or octonions) (dimension 8) have
many powerful applications. See the recent book "On Quaternions and
Octonions" by Conway and Smith for more information. There have been
articles by John Baez discussing physical applications, too.

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