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  "body": "\n## Statement\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $f \\in K[x]$ have degree $n \\ge 1$.\nThen $f$ has a [[splitting field|02NZZ1J0FH]] $L$ over $K$, and $[L : K] \\le n!$.\n\n## Remarks\n\nThe proof is by induction on $n$. [[Some extension $K_1$ contains a root $a_1$ of $f$|0VG02M1FEZ]], with $[K_1 : K] \\le n$. Then $f = (x - a_1) g$ in $K_1[x]$, and a splitting field of $g$ over $K_1$ is a splitting field of $f$ over $K$.\nThe bound $n!$ follows from the [[tower law|0A7KJ68XHT]].\nThe bound is reached: the splitting field of $x^3 - 2$ over $\\mathbb{Q}$ has degree $6 = 3!$.\n\n## Explanations\n\n### L1 human\n\nYou build a splitting field one root at a time.\nEach step adds a root of a remaining factor, and the degree of the step is at most the degree of that factor.\n",
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