Bott-Milnor-Kervaire 1,2,4,8 Theorem

Theorem
Let $A$ be a division algebra with real scalars.

Then the dimension of $A$ is either:
 * $1$: the real numbers $\R$
 * $2$: the complex numbers $\C$
 * $4$: the quaternions $\Bbb H$

or:
 * $8$: the octonions $\Bbb O$.