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  "body": "\n## Statement\n\nLet $D$ be a [[division ring|0WZXK05ME7]] with a finite number of elements.\nThen the multiplication of $D$ is commutative, so $D$ is a [[field|0QA8CHH8MG]].\n\n## Remarks\n\nThe hypothesis of finiteness is necessary: the [[quaternions|0GE4DVQKR8]] are an infinite division ring that is not commutative.\nA consequence is that every finite domain, commutative or not, is a field.\n\n## History\n\nJoseph Wedderburn gave the first proof in 1905.\nLeonard Eugene Dickson gave another proof soon after.\nThe first proof of Wedderburn had a gap, and for this reason some historians credit Dickson with the first correct proof.\n\n## Explanations\n\n### L1 human\n\nIn an infinite division ring, $a b$ and $b a$ can be different.\nThe theorem says that this cannot happen when the ring is finite.\nSo there are no finite analogues of the quaternions.\n",
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