Hilbert's theorem 90

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This fact is related to: group cohomology
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Let L/K be a Galois extension of fields and let G be the Galois group of the field extension. If G is a cyclic group, then for the usual action of G on L^*, the first cohomology group is trivial. In other words:

H^1(G,L^*) = 0

In other words, any 1-cocycle for the action on L^* is in fact a 1-coboundary.