### Frobenius theorem on real division algebras

This post is a proof of the well known Frobenius theorem. It’s nothing groundbreaking, I’m just writing it down to organize my own thoughts.

**Definition.** A *real division algebra* is a set together with operations , and such that:

- is a vector space over .
- is a division ring.
- for every and .

Examples of real division algebras: itself, the algebra of complex numbers , the algebra of quaternions .

**Theorem** (Frobenius)**.** Every finite-dimensional real division algebra is isomorphic to one of , or .