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I know that the only division algebras over the real numbers have dimension $1, 2, 4,$ and $8$ (real numbers, complex numbers, quaternions, octonions). I also know that those are the only numbers of squares for which a bilinear generalization of Euler's four square identity exists (product of squares, Brahmagupta-Fibonacci identity, Euler's four square identity, Degen's eight square identity). Are these two facts related in any way or is this just a coincidence? I feel that it might just be a coincidence because the only dimension for an algebra over algebraically closed fields is 1 and there are infinitely many dimensions for division algebras over fields which aren't algebraically closed or real closed.

mathlander
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  • Related: https://math.stackexchange.com/questions/2892818/finite-dimensional-division-algebras-over-the-reals-other-than-mathbbr-math – mathlander Oct 25 '22 at 00:32
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    The square identities are just $|xy|^2=|x|^2|y|^2$ in the corresponding composition algebra. – anon Oct 25 '22 at 00:38

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