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  "body": "\n## Summary\n\nInfinite Galois theory extends [[Galois theory|WNQDKQYZ2R]] to [[Galois extensions|0DS2F7GK05]] of infinite degree, such as the algebraic closure of $\\Q$ over $\\Q$.\nThe Galois group then carries the [[Krull topology|F3GTF5NT4M]], which makes it a [[profinite group|KA0YDB6QXE]].\nThe fundamental theorem survives with one change: intermediate fields correspond to the closed subgroups, not to all subgroups.\n\n## Explanations\n\n### L1 human\n\nAn infinite Galois extension is the union of its finite Galois pieces.\nEach symmetry of the whole extension is a compatible choice of symmetries of all these pieces.\nTwo symmetries are close in the Krull topology when they agree on a large finite piece.\nFor example, the symmetries of the algebraic closure of $\\F_p$ include the Frobenius map $x \\mapsto x^p$, and the closed subgroup it generates is the whole group.\n",
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