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  "body": "\n## Summary\n\nGrothendieck's Galois theory recasts [[Galois theory|WNQDKQYZ2R]] as an equivalence of categories.\nFor a field $k$, it matches finite étale $k$-algebras with finite sets on which the [[absolute Galois group|PGKWBJDTKX]] of $k$ acts continuously.\nThe same pattern, with Galois categories and fibre functors, defines the étale fundamental group of a scheme.\nIt thus puts Galois groups of fields and fundamental groups of spaces in one frame.\n\n## Explanations\n\n### L1 human\n\nClassical Galois theory matches intermediate fields with subgroups of a Galois group.\nGrothendieck's view looks instead at all finite separable algebras at once, such as products of fields.\nEach such algebra is replaced by the finite set of its embeddings into a separable closure, and the Galois group permutes this set.\nFor $k = \\R$, the algebra $\\C$ gives a set of two points swapped by complex conjugation, while $\\R \\times \\R$ gives two points that stay fixed.\nThe picture is the same as for covering spaces, where the fundamental group permutes the points of a fibre.\n",
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