= Galois cohomology
{c}
{title2=$H^i(G,M)$}
{wiki}
<Galois cohomology> is continuous <group cohomology> of a <Galois group> acting on a discrete <Galois module>. For a field $K$, the usual group is its <absolute Galois group> $G_K=\operatorname{Gal}(K_s/K)$. In degree zero it gives the invariant subgroup. In degree one, continuous cocycles satisfy
$$
c(\sigma\tau)=c(\sigma)+\sigma c(\tau),
$$
and two cocycles have the same class when their difference is $\sigma\mapsto\sigma m-m$. If the action is trivial, these are continuous homomorphisms modulo the zero subgroup of coboundaries.
Continuity matters for the profinite group $G_K$: it connects the cohomology to finite Galois subextensions. <Hilbert's theorem 90> makes the first multiplicative cohomology of a finite Galois extension vanish, and the Kummer exact sequence turns that vanishing into the power-class description in <Kummer theory>.
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