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, Corollary 4.4.3. I assume that this is connected, perhaps even implies the main theorem from Kummer theory stating that, if $K$ has a primitive $n$th root of unity, then there is a correspondence between abelian extensions of exponent $n$ and subgroups of $K^{\times}/(K^{\times})^n$. I assume this is because $H^1\_{et}(\mathrm{Spec}(K),\mu\_n) = H^1\_{et}(\mathrm{Spec}(K),\mathbb{Z}/n\mathbb{Z})$ is the Galois group of the maximal abelian extension of exponent $n$ of $K$? If so, can you give me a reference for this result? Sorry if this question is too basic, I didn't touch etale cohomology for years.



Etale cohomology and Kummer theory
The question was answered by R. van Dobben de Bruyn in the comments.



> 
> "Serre's Galois cohomology states this relation briefly in a remark
> (p. 73 in §II.1.2 of the second English edition), and refers to
> Bourbaki's Algèbre (Chap. V) for a statement of Kummer theory (but
> Bourbaki doesn't use Galois cohomology). In short: yes, this does
> imply Kummer theory with very little work."
> 
> 
> 


