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  "body": "\n## Summary\n\nGalois cohomology is the group cohomology of Galois groups, see the [[definition|EESNTXFWHZ]].\nThe main case is the [[absolute Galois group|PGKWBJDTKX]] of a field, acting continuously on a module.\nMany arithmetic invariants of a field are Galois cohomology groups.\n[[Hilbert's Theorem 90|YFF8B5A0QG]] says that $H^1$ of the multiplicative group vanishes.\n[[Kummer theory|XZQENYV15Y]] computes $H^1$ of the roots of unity, and $H^2$ of the multiplicative group is the [[Brauer group|BEW48TA9WZ]].\nIt is a central language of algebraic number theory, class field theory and arithmetic geometry.\n\n## Explanations\n\n### L1 human\n\nWhen a group of symmetries acts on some objects, cohomology measures the ways in which \"local\" choices fail to fit together into one symmetric choice.\nFor a field $K$, the symmetries are the automorphisms of a large extension field that fix $K$.\nThe group $H^0$ is just the set of objects fixed by all symmetries.\nThe group $H^1$ counts, for example, the different forms over $K$ of an object that become the same over the large field.\nFor instance, over $\\Q$ the field extensions of degree $2$ correspond to the nonzero elements of $H^1$ with coefficients $\\set{\\pm 1}$.\n",
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