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  "body": "\n## Definition\n\nLet $L/K$ be a [[Galois extension|0DS2F7GK05]], that is, an [[algebraic extension|09EYMKMSB9]] that is normal and separable, of possibly infinite degree.\n*Infinite Galois theory* is the study of $L/K$ through its [[Galois group|0XVR72M51M]] $G = \\Gal(L/K)$ with the [[Krull topology|F3GTF5NT4M]].\nIts central statement is the Galois correspondence:\n\n1. the maps $H \\mapsto L^H$, with $L^H$ the [[fixed field|0KMJ6919WS]] of $H$, and $E \\mapsto \\Gal(L/E)$ are inverse bijections that reverse inclusion.\n   They match the closed subgroups $H$ of $G$ with the [[intermediate fields|02NBKVQFRQ]] $E$ of $L/K$;\n2. a closed subgroup $H$ is open if and only if $L^H/K$ is finite;\n3. a closed subgroup $H$ is normal if and only if $L^H/K$ is Galois, and then $\\Gal(L^H/K) \\cong G/H$ as topological groups.\n\n## Remarks\n\nExample: let $K = \\F_p$ and $L$ an algebraic closure of $\\F_p$.\nThe intermediate fields are the fields $\\F_{p^n}$ and the infinite unions of such fields, and $\\F_{p^n}$ corresponds to the open subgroup generated topologically by $\\phi^n$, where $\\phi(x) = x^p$.\nNon-example of the correspondence for all subgroups: the subgroup $H = \\set{\\phi^k : k \\in \\Z}$ is not closed in $\\Gal(L/\\F_p)$.\nIts fixed field is $\\F_p$, the same as for the whole group, so $H \\mapsto L^H$ is not injective on all subgroups.\nWhen $L/K$ is finite, the Krull topology is discrete, every subgroup is closed, and the theory reduces to the [[fundamental theorem of Galois theory|0T1Q8D0FW8]].\n",
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