Frobenius theorem on real division algebras

Remarks

Proof idea: each $a \in D$ lies in the commutative subalgebra $\R[a]$, which is a finite field extension of $\R$, hence $\R$ or $\C$. Using this, one shows that the set $V$ of elements $u$ with $u^2 \in \R$ and $u^2 \le 0$ is a real subspace with $D = \R \oplus V$. The rule $\ip{u}{w} = -(uw + wu)/2$ is a positive definite inner product on $V$. An orthonormal basis of $V$ satisfies the multiplication rules of $\C$ or $\mathbb{H}$, and a fourth basis vector would lead to a contradiction. The dimensions are $1$, $2$ and $4$. Associativity is needed: the octonions form an $8$-dimensional real algebra, not associative, in which every nonzero element has an inverse. Finite dimension is needed: the field $\R(t)$ of rational functions is a commutative division algebra over $\R$ of infinite dimension.

Going deeper

For a student

In the language of the Brauer group, the only central division algebras over $\R$ are $\R$ and $\mathbb{H}$. So the Brauer group of $\R$ has order $2$. This agrees with Galois cohomology: $\C$ is the algebraic closure of $\R$, and $H^2(\Gal(\C/\R), \C^\times) \cong \Z/2\Z$. Without associativity, a finite-dimensional real division algebra has dimension $1$, $2$, $4$ or $8$, by work of Kervaire and of Bott and Milnor. Hurwitz's theorem shows that the normed ones are $\R$, $\C$, $\mathbb{H}$ and the octonions.