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}.

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