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  "body": "\n## Statement\n\nLet $R$ be an [[integral domain|09S3PFBEPM]] that has a finite number of elements.\nThen $R$ is a [[field|0QA8CHH8MG]].\n\n## Remarks\n\nThe proof: for $a \\neq 0$, the map $x \\mapsto a x$ from $R$ to $R$ is injective by [[cancellation|0GJM24EXP9]]. A finite set has no injective map to itself that is not surjective, so $a x = 1$ for some $x$.\nA consequence is that [[$\\mathbb{Z}/p\\mathbb{Z}$ is a field|0NV6E43P4A]] for every prime $p$.\nThe hypothesis of finiteness is necessary: $\\mathbb{Z}$ is an infinite integral domain that is not a field.\n\n## Explanations\n\n### L1 human\n\nIn a finite integral domain, multiplication by a nonzero element $a$ never sends two elements to the same element.\nSince there are only finitely many elements, every element is a product $a x$.\nIn particular, $1 = a x$ for some $x$, and $x$ is the inverse of $a$.\n",
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