Hilbert's theorem 90

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.