Hurwitz's theorem on composition algebras

Remarks

The idea of the proof: let $B$ be a proper unital subalgebra of $A$ on which the norm is nondegenerate. Choose $v$ orthogonal to $B$ with $N(v) \neq 0$. Then $B + Bv$ is a subalgebra of twice the dimension, isomorphic to the Cayley–Dickson double of $B$ with parameter $-N(v)$. The double of $B$ is again a composition algebra only when $B$ is associative. The double is associative only when $B$ is commutative and associative. So the doubling from $K$ stops after at most three steps. Example: over $\R$, the composition law for $\mathbb{H}$ gives Euler's four-square identity, and the one for $\mathbb{O}$ gives Degen's eight-square identity.

Going deeper

For a student

The theorem classifies the normed division algebras: they are the composition algebras over $\R$ with a positive definite norm. It also answers the question of Hurwitz about sum-of-squares identities: an identity $(x_1^2 + \cdots + x_n^2)(y_1^2 + \cdots + y_n^2) = z_1^2 + \cdots + z_n^2$ with each $z_k$ bilinear in $x$ and $y$ exists only for $n = 1, 2, 4, 8$. Over $\R$, $\C$ and finite fields, the classification becomes a classification of quadratic forms, since the algebra is determined by its norm.