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  "body": "\n## Statement\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $f \\in K[x]$ be a nonzero polynomial of [[degree|0DPF6WRZW6]] $n$.\nThen $f$ has at most $n$ roots in $K$.\nMore precisely, the sum of the [[multiplicities|03XVP8Q4GY]] of the roots of $f$ in $K$ is at most $n$.\n\n## Remarks\n\nThe proof is an induction on $n$ with the [[factor theorem|0K3MWA7DMC]]: a root $c$ gives $f = (x - c) g$ with $\\deg g = n - 1$.\nThe same statement holds over every [[integral domain|09S3PFBEPM]].\nIt fails without this hypothesis: in $\\mathbb{Z}/8\\mathbb{Z}$, the polynomial $x^2 - 1$ has the four roots $1, 3, 5, 7$.\n\n## Explanations\n\n### L1 human\n\nA polynomial of degree $2$ over the real numbers has at most two roots.\nIn general, each root takes away one linear factor, and you can do this at most $n$ times.\nStrange rings like $\\mathbb{Z}/8\\mathbb{Z}$ can break this rule, because they have zero divisors.\n",
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