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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $K^{s}$ be a [[separable closure|JJ91A1V8NQ]] of $K$.\nThe *absolute Galois group* of $K$ is the [[Galois group|0XVR72M51M]]\n$$G_K = \\Gal(K^{s}/K)$$\nof field automorphisms of $K^{s}$ that fix $K$ pointwise, with the [[Krull topology|F3GTF5NT4M]].\n\n## Remarks\n\nIt does not depend on the choice of $K^{s}$ up to isomorphism, because two separable closures are isomorphic over $K$.\nThe induced isomorphism of Galois groups is unique only up to an inner automorphism.\nIf $K$ is perfect, for example of characteristic $0$, then $K^{s}$ is an algebraic closure of $K$.\nExample: $G_\\R = \\Gal(\\C/\\R)$ has order $2$, generated by complex conjugation, since $\\C$ is algebraically closed and $[\\C : \\R] = 2$.\nExample: for a finite field $\\F_q$, the group $G_{\\F_q}$ is isomorphic to the profinite completion $\\hat{\\Z}$ of $\\Z$, topologically generated by $x \\mapsto x^q$.\nNon-example: $\\Gal(\\Q(i)/\\Q)$ is not the absolute Galois group of $\\Q$, since $\\Q(i)$ is not separably closed; $x^2 - 2$ has no root in it.\nIf $K$ is separably closed, then $G_K$ is trivial.\n",
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