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  "body": "\n## Definition\n\nLet $K$ be a field, $K^s$ a [[separable closure|JJ91A1V8NQ]] of $K$, and $G_K = \\Gal(K^s/K)$ the [[absolute Galois group|PGKWBJDTKX]] with its Krull topology, a [[profinite group|KA0YDB6QXE]].\nA *discrete $G_K$-module* is an [[abelian group|071DDFVG4J]] $M$ with a [[group action|0YSE802YCQ]] of $G_K$ by group automorphisms.\nThe stabilizer of each $m \\in M$ must be an open subgroup of $G_K$.\n\nFor $n \\ge 0$, let $C^n(G_K, M)$ be the group of continuous maps $G_K^n \\to M$, where $M$ has the discrete topology and $C^0(G_K, M) = M$.\nDefine $d \\colon C^n(G_K, M) \\to C^{n+1}(G_K, M)$ by\n$$(d\\varphi)(g_1, \\ldots, g_{n+1}) = g_1 \\varphi(g_2, \\ldots, g_{n+1}) + \\sum_{i=1}^{n} (-1)^i \\varphi(g_1, \\ldots, g_i g_{i+1}, \\ldots, g_{n+1}) + (-1)^{n+1} \\varphi(g_1, \\ldots, g_n).$$\nThen $d \\circ d = 0$.\nThe *Galois cohomology groups* of $K$ with coefficients in $M$ are\n$$H^n(K, M) = H^n(G_K, M) = \\ker\\big(d \\colon C^n \\to C^{n+1}\\big) / \\im\\big(d \\colon C^{n-1} \\to C^n\\big),$$\nwith $C^{-1} = 0$.\n\n## Remarks\n\nThe group $H^0(K, M)$ is the group $M^{G_K}$ of fixed elements.\nThe group $H^n(K, M)$ is also the direct limit, over finite Galois extensions $L/K$ inside $K^s$, of the group cohomology groups $H^n(\\Gal(L/K), M^{\\Gal(K^s/L)})$.\nExample: $M = (K^s)^\\times$ is a discrete $G_K$-module, since each element of $K^s$ lies in a finite extension of $K$, and its stabilizer contains the open subgroup fixing that extension.\nFor it, $H^0(K, (K^s)^\\times) = K^\\times$, $H^1(K, (K^s)^\\times) = 0$ by [[Hilbert's Theorem 90|YFF8B5A0QG]], and $H^2(K, (K^s)^\\times)$ is the [[Brauer group|BEW48TA9WZ]] of $K$.\nFor an integer $m$ not divisible by the characteristic of $K$, with $\\mu_m$ the group of $m$-th roots of unity in $K^s$, [[Kummer theory|XZQENYV15Y]] gives $H^1(K, \\mu_m) \\cong K^\\times / (K^\\times)^m$.\nNon-example: let $\\ell$ be a prime and $\\Z_\\ell(1)$ the inverse limit of the groups $\\mu_{\\ell^n}$ in $\\overline{\\Q}$, with $G_\\Q$ acting on each $\\mu_{\\ell^n}$.\nWith the discrete topology, $\\Z_\\ell(1)$ is not a discrete $G_\\Q$-module.\nThe stabilizer of a generator is the subgroup fixing all $\\ell$-power roots of unity.\nIt is closed but not open, because the field of all $\\ell$-power roots of unity has infinite degree over $\\Q$.\nFor such modules one uses continuous cochains for the $\\ell$-adic topology instead.\n",
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