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