TITLE: Why are the only division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?
QUESTION [55 upvotes]: Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?
Here a division algebra is an associative algebra where every nonzero number is invertible (like a field, but without assuming commutativity of multiplication).
This is an old result proved by Frobenius, but I can't remember how the argument goes.  Anyone have a quick proof?

REPLY [2 votes]: There is another proof using the theory of central simple algebras and quaternion algebras.
Denote by $D$ such a division algebra. Note that a division algebra is always simple. Moreover, since the center of a division algebra is a field, $D$ is central simple over either $\mathbb{C}$ or $\mathbb{R}$.
As there is no nontrivial division algebra over an algebraically closed field, $D$ must be $\mathbb{C}$ if $Z(D)=\mathbb{C}$.
Otherwise, $D$ is central simple over $\mathbb{R}$, then $\mathbb{C}$ splits $D$. Hence we obtain the following:
$$
\sqrt{\mathbf{dim}_{\mathbb{R}}(D)} = \mathbf{ind}_{\mathbb{R}}(D)|[\mathbb{C}:\mathbb{R}]=2
$$
$D$ is nontrivial, hence the dimension of $D$ is 4. Then $D$ must be $\mathbb{H}$ by Wedderburn's theorem.