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  "body": "\n## Statement\n\nLet $K$ be a [[field|0QA8CHH8MG]], and let $L$ and $L'$ be [[algebraic closures|0MCXX29HWY]] of $K$.\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 uses Zorn's lemma on the partial isomorphisms between subfields of $L$ and $L'$ that fix $K$.\nThe isomorphism is not unique in general. For example, complex conjugation and the identity are two such isomorphisms from $\\mathbb{C}$ to $\\mathbb{C}$ over $\\mathbb{R}$.\nSo one often writes \"the\" algebraic closure $\\overline{K}$.\nThe statement is an infinite form of the [[uniqueness of splitting fields|069A3BC3EP]].\n\n## Explanations\n\n### L1 human\n\nTwo algebraic closures of the same field are the same field with different names for the elements.\nThere can be many ways to match them, so the matching is not part of the structure.\n",
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