Hilbert's theorem 90

This fact is related to: group cohomology
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Statement

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.