### 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 $A$ together with operations $+: A \times A \to A$, $\cdot : \mathbb {R}\times A \to A$ and $\ast : A \times A \to A$ such that:

1. $(A, +, \cdot )$ is a vector space over $\mathbb {R}$.
2. $(A, +, \ast )$ is a division ring.
3. $r \cdot (a_1 \ast a_2) = (r \cdot a_1) \ast a_2 = a_1 \ast (r \cdot a_1)$ for every $r \in \mathbb {R}$ and $a_1, a_2 \in A$.

Examples of real division algebras: $\mathbb {R}$ itself, the algebra of complex numbers $\mathbb {C}$, the algebra of quaternions $\mathbb {H}$.

Theorem (Frobenius). Every finite-dimensional real division algebra is isomorphic to one of $\mathbb {R}$, $\mathbb {C}$ or $\mathbb {H}$.