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  "body": "\n## Statement\n\nLet $K$ be a [[field|0QA8CHH8MG]], let $f \\in K[x]$ be nonzero, and let $L$ and $L'$ be [[splitting fields|02NZZ1J0FH]] of $f$ over $K$.\nThen there is a ring isomorphism $\\sigma \\colon L \\to L'$ with $\\sigma(c) = c$ for every $c \\in K$.\n\n## Remarks\n\nThe proof is by induction on $[L : K]$. It extends an isomorphism one root at a time, with the [[structure of simple extensions|0ZN39QDTP9]].\nThe isomorphism $\\sigma$ is usually not unique. The isomorphisms from $L$ to itself that fix $K$ form the [[Galois group|0XVR72M51M]] of $f$.\nSo one speaks of \"the\" splitting field of $f$ over $K$.\nThe choice of the irreducible factors in the construction does not change the result, up to isomorphism.\n\n## Explanations\n\n### L1 human\n\nTwo splitting fields of the same polynomial are the same field with different names for the elements.\nThe proof extends a matching of the base field, one root at a time.\n",
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