The Name for $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ Exclusively Algebras?

Question

The Wikipedia page for Normed Division Algebras has been redirected to Normed Algebras and the explanation given is that $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ algebras are not the only normed division algebras.

What is the name for the class of algebras consisting of $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ exclusively?

Answer 1

I would call them the real composition algebras. The result that states they are the only four is called Hurwitz's theorem, and in Jacobson's Basic algebra I you can find the generalized version which applies to fields beyond $\Bbb R$.

The definition of a composition algebra is basically "a nonassociative unital algebra with a nondegenerate bilinar form making a norm." I thought I remember elsewhere some characterization in terms of an involution too, but I'm sticking with Jacobson's definition for now.

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Please note also that the comment you linked to about the theorem being "incorrect" is not really anything to pay attention to. The author takes a minority stance that the inclusion of identity is not assumed. However, it is assumed for all statements of the Hurwitz theorem that I've seen. There is no shortage of such comments on wikipedia, which is why you have to be careful what you believe :)

Answer 2

This list of algebras stems from the Frobenius theorem for finite division algebras over the real numbers. As one goes further down the list, properties of the real numbers are lost along the way.

1. Real numbers: totally ordered, commutative, associative
2. Complex numbers: commutative, associative
3. Quaternions: associative
4. Octonions: none of the above

Answer 3

Euclidean Hurwitz algebras

"Theorem. The only Euclidean Hurwitz algebras are the real numbers, the complex numbers, the quaternions and the octonions."