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  "body": "\n## Statement\n\nLet $K$ and $L$ be [[algebraically closed fields|02SJG1HC32]], and let $P_K$ and $P_L$ be their prime subfields.\nThen $K$ and $L$ are isomorphic if and only if they have the same [[characteristic|1ZB13N78XW]] and $\\operatorname{trdeg}_{P_K} K = \\operatorname{trdeg}_{P_L} L$, where $\\operatorname{trdeg}$ is the [[transcendence degree|D7TR0YWR9H]].\n\n## Remarks\n\nIf the characteristics agree, then $P_K \\cong P_L$, and a bijection between [[transcendence bases|E45PA267S4]] $B$ of $K$ and $B'$ of $L$ gives an isomorphism $P_K(B) \\cong P_L(B')$.\nSince $K$ and $L$ are [[algebraic closures|0MCXX29HWY]] of these fields, the isomorphism extends to $K \\cong L$ by [[uniqueness of the algebraic closure|01VQ7C4FTB]].\nThe converse holds because an isomorphism preserves the characteristic and maps a transcendence basis to a transcendence basis.\nFor an uncountable algebraically closed field, the transcendence degree over the prime field equals the [[cardinality|0JTGRV4WT4]] of the field.\nSo two uncountable algebraically closed fields of the same characteristic and the same cardinality are isomorphic; for example, $\\C$ is isomorphic to an algebraic closure of $\\Q(x_i : i \\in \\R)$.\n\n## Explanations\n\n### L1 human\n\nAn algebraically closed field is determined by two numbers: its characteristic and how many free variables it has.\nEverything else follows by adding all roots of all polynomials.\nFor instance, $\\C$ and the algebraic closure of the field of $p$-adic numbers are isomorphic as fields, although their topologies are very different.\n\n### L2 human\n\nThe theorem shows that the algebraic closure $\\overline{\\Q}$ and $\\C$ are not isomorphic: one has transcendence degree $0$, the other has transcendence degree the continuum.\nIt is the field-theoretic source of the fact that the theory of algebraically closed fields of a fixed characteristic is categorical in each uncountable cardinality.\nIts proof uses the existence of transcendence bases and the uniqueness of algebraic closures, both of which rest on Zorn's lemma.\n",
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