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  "body": "\n## Statement\n\nLet $D$ be a finite-dimensional associative algebra over the field $\\R$ of [[real numbers|08J2M1JYAG]].\nSuppose $D$ is a [[division ring|0WZXK05ME7]]: it has a unit $1 \\ne 0$ and every nonzero element has a two-sided inverse.\nThen $D$ is isomorphic, as an $\\R$-algebra, to exactly one of $\\R$, the [[complex numbers|09FTAET5Z4]] $\\C$, and the [[quaternions|0GE4DVQKR8]] $\\mathbb{H}$.\n\n## Remarks\n\nProof idea: each $a \\in D$ lies in the commutative subalgebra $\\R[a]$, which is a finite field extension of $\\R$, hence $\\R$ or $\\C$.\nUsing 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 $D = \\R \\oplus V$.\nThe rule $\\ip{u}{w} = -(uw + wu)/2$ is a positive definite inner product on $V$.\nAn orthonormal basis of $V$ satisfies the multiplication rules of $\\C$ or $\\mathbb{H}$, and a fourth basis vector would lead to a contradiction.\nThe dimensions are $1$, $2$ and $4$.\nAssociativity is needed: the octonions form an $8$-dimensional real algebra, not associative, in which every nonzero element has an inverse.\nFinite dimension is needed: the field $\\R(t)$ of rational functions is a commutative division algebra over $\\R$ of infinite dimension.\n\n## Explanations\n\n### L1 human\n\nConsider number systems of finite dimension over the reals, where one can divide by every nonzero number.\nIf the order of brackets in a product does not matter, there are only three of them.\nThey are the real numbers, the complex numbers and Hamilton's quaternions.\nThe quaternions are the only one of the three in which $ab$ and $ba$ can differ.\n\n### L2 human\n\nIn the language of the [[Brauer group|BEW48TA9WZ]], the only central division algebras over $\\R$ are $\\R$ and $\\mathbb{H}$.\nSo the Brauer group of $\\R$ has order $2$.\nThis agrees with [[Galois cohomology|9MPCZ3EAJM]]: $\\C$ is the [[algebraic closure|0X6AB5P3G1]] of $\\R$, and $H^2(\\Gal(\\C/\\R), \\C^\\times) \\cong \\Z/2\\Z$.\nWithout associativity, a finite-dimensional real division algebra has dimension $1$, $2$, $4$ or $8$, by work of Kervaire and of Bott and Milnor.\nHurwitz's theorem shows that the normed ones are $\\R$, $\\C$, $\\mathbb{H}$ and the octonions.\n",
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