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It is a famous theorem by Hurwitz that there are only four composition algebras (more exactly, Euclidean Hurwitz algebras): the real numbers, the complex numbers, the quaternions, and the octonions.

But why aren't there five? It would seem that the trivial algebra $\left\{ 0 \right\}$ fits the bill.

Here's the definition I'm using for now: http://en.wikipedia.org/wiki/Composition_algebra

Perhaps it's because a Hurwitz algebra is demanded to be unital, but why not let $1 = 0$ in this case? Perhaps it's because people want to say that all the composition algebras are generated by the smallest such algebra, which wouldn't be true if we included the trivial algebra. But that seems kind of arbitrary to me; if anything, just as arbitrary as including the trivial algebra.

Is there anything deeper? Thanks for the thoughts.

Doug
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  • So ... this question doesn't really have anything to do with Hurwitz theorem? If so, then why reference it at all? Wikipedia speaks of "composition algebra over a field": if one is willing to accept this as natural, the question then becomes why we don't allow the trivial ring to be a field, which is answered elsewhere (it is not just arbitrary, and the trivial ring in fact wouldn't solve any Field With One Element questions for instance), see for instance this question. – anon Mar 21 '14 at 14:40
  • The Hurwitz theorem is part of his motivation for asking this question... – Bulberage Mar 21 '14 at 14:47
  • The underlying field in the context of my question is the real numbers. Hurwitz' Theorem regards "Euclidean Hurwtiz algebras", which are unital composition algebras over the real numbers where the quadratic form is positive definite. Perhaps I should have been more explicit. – Doug Mar 21 '14 at 14:51
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    In the usual definition one requires $1\ne0$, so the base field embeds in the algebra. It's nothing more than a terminology question. – egreg Mar 21 '14 at 14:57
  • There is infinitely many finite-dimensional composition algebras over R. But only 5 Hurwits algebras. So, what you are asking about? – Anixx Mar 19 '21 at 13:23

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I will accept egreg's comment as the answer to this question.

Doug
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