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If $K$ is a field and $n \geq 1$ is such that $n \in K^{\times}$, then $H^1_{et}(\mathrm{Spec}(K),\mu_n)=K^{\times} / (K^{\times})^n$. This is easy to prove, see for instance Tamme, Etale Cohomology, ...

I posted this question over on stack exchange and was told it would work better here. For interest I have been looking at links between class field theory and étale cohomology. Let $k$ be a global ...