= Finite-dimensional real division-algebra classification
{title2=$D\cong\mathbb R,\mathbb C,\mathbb H$}
= Frobenius theorem on real division algebras
{c}
{synonym}
A finite-dimensional associative <division algebra> over $\mathbb R$ is one of the three displayed algebras. A nonreal element has an irreducible quadratic <minimal polynomial> and yields $i^2=-1$. Its centralizer is a <division algebra> over central $\mathbb C$, hence is $\mathbb C$ by splitting <minimal polynomials>. Decompose the whole algebra into the plus and minus eigenspaces of conjugation by $i$. If the minus space is zero, the algebra is $\mathbb C$; otherwise it is $\mathbb Cj$, since multiplying by $j^{-1}$ takes that space into the centralizer. One has $jz=\overline z j$ and $j^2\in\mathbb R$. Division forces $j^2<0$, which can be rescaled to $-1$, yielding $\mathbb H$. Of these, only $\mathbb R,\mathbb H$ have center $\mathbb R$.
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