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  "body": "\n## Definition\n\nLet $L/K$ be a [[Galois extension|0DS2F7GK05]], possibly of infinite degree, with [[Galois group|0XVR72M51M]] $G = \\Gal(L/K)$.\nLet $\\mathcal{E}$ be the set of [[intermediate fields|02NBKVQFRQ]] $E$ of $L/K$ with $[E : K]$ finite.\nThe *Krull topology* on $G$ is the following [[topology|0CNQXVBHCX]].\nA subset $U \\subseteq G$ is open if and only if for each $\\sigma \\in U$ there is $E \\in \\mathcal{E}$ with $\\sigma \\Gal(L/E) \\subseteq U$.\nEquivalently, the cosets $\\sigma \\Gal(L/E)$, for $\\sigma \\in G$ and $E \\in \\mathcal{E}$, form a basis of the topology.\n\n## Remarks\n\nWith this topology $G$ is a topological group that is [[compact|0QPHW703PV]], [[Hausdorff|0MXB4CKHFW]] and totally disconnected, that is, a [[profinite group|KA0YDB6QXE]].\nIn fact $G$ is isomorphic, as a topological group, to the inverse limit of the finite groups $\\Gal(E/K)$ over the finite Galois subextensions $E/K$.\nExample: if $L/K$ is finite, then $L \\in \\mathcal{E}$ and $\\Gal(L/L) = \\set{\\id}$, so every subset of $G$ is open and the topology is discrete.\nNon-example of an open set: for $L$ an algebraic closure of $\\F_p$, the one-point set $\\set{\\id}$ is not open.\nIndeed it would contain some $\\Gal(L/\\F_{p^n})$, which is infinite since $L/\\F_{p^n}$ has infinite degree.\n",
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