If $\phi:G\longrightarrow A^{\times}$ is a nontrivial character, why is $H^1(G,A)=0$?

Question

If $\phi:G\longrightarrow A^{\times}$ is a nontrivial character and $M=A^1$, where $A$ is a ring, why is $H^1(G,M)=0$ if $G$ acts via $\phi$?

This is essentially the claim that Wiles makes on p. 465 of his Fermat paper, in the course of the proof of Prop. 1.2.