Let $k$ be a number field. We have the well-known Kummer exact sequence of etale sheaves on $\mathrm{Spec}\, k$: $$1 \rightarrow \mu_n \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 1.$$

**Question 0.** Applying the etale cohomology functor $H_{et}^i(k,-)$, I know that $H_{et}^i(k,\mathbb{G}_m) = H^i(k,\bar{k}^*)$, where the latter is a Galois cohomology group. What is the Galois cohomological equivalent for $H^i_{et}(k,\mu_n)$?

Let $\mu_\infty := \mathrm{colim}_n\mu_n$, this group can be interpreted as the set of all $n$-th roots of unity, i.e., $\mu_\infty \cong \mathbb{Q}/\mathbb{Z}$. We work only in Galois cohomology.

**Question 1.** I would like to know if we could obtain a similar Kummer sequence involving $\mu_\infty$ instead of $\mu_n$ for a fixed $n$. The reason for this is that I came across an exact sequence $$0 \rightarrow \mathrm{Br}(k) \otimes _\mathbb{Z} \mathbb{Q}/\mathbb{Z} \rightarrow H^3(k,\mu_\infty) \rightarrow H^3(k,\bar{k}^*) \rightarrow 0$$ but I am unable to derive it from the usual Kummer sequence.

**Question 2.** The idea for this exact sequence is to prove that the middle term is 0. It is known that we have $H^3(k,\bar{k}^*)=0$ in our setting and for any non-archimedean place $v$, we have $\mathrm{Br}(k_v) \subset \mathbb{Q}/\mathbb{Z}$. But I'm also unsure how does it imply that $\mathrm{Br}(k) \otimes _\mathbb{Z} \mathbb{Q}/\mathbb{Z}= 0$.