State Frobenius’s Theorem classifying all finite-dimensional associative division algebras over the real numbers \(\mathbb{R}\), and state the generalized Hurwitz / Adams theorem for normed / alternative division algebras.
Solution
There are three related statements that should be distinguished.
Frobenius’ theorem. The finite-dimensional associative division algebras over \(\mathbb R\) are exactly \[ \boxed{\mathbb R,\ \mathbb C,\ \mathbb H}. \] Their real dimensions are \(1,2,4\).
Hurwitz’ theorem. The finite-dimensional real normed division algebras, equivalently positive-definite real composition division algebras, are exactly \[ \boxed{\mathbb R,\ \mathbb C,\ \mathbb H,\ \mathbb O}, \] of dimensions \(1,2,4,8\). The octonions \(\mathbb O\) are nonassociative but alternative.
For more general finite-dimensional real division algebras, one no longer has uniqueness up to isomorphism in dimensions \(4\) and \(8\). What survives is the dimension restriction: the existence of a finite-dimensional real division algebra forces \[ \boxed{\dim_\mathbb R D\in\{1,2,4,8\}}, \] a consequence of the Hopf-invariant-one/division-algebra results of Bott–Milnor–Kervaire and Adams.
Thus Frobenius is an isomorphism classification in the associative case; Hurwitz is an isomorphism classification in the normed composition case; and the general topological theorem is a dimension restriction, not an isomorphism classification.