Let $F/K$ be a Galois extension of number fields with Galois group $G$. Let $\mathcal{O}\_F$ and $\mathcal{O}\_K$ be the associated rings of integers, and let $n\geq 1$.



> 
> When is
>  $$
> K\_{2n-1}(\mathcal{O}\_F)^G \cong K\_{2n-1}(\mathcal{O}\_K)?
> $$
> 
> 
> 


It would be good enough for me to have this on the prime-to-2 parts. So let $p$ be an odd prime. Then
$$
K\_{2n-1}(\mathcal{O}\_F)\otimes\_{\mathbb{Z}}\mathbb{Z}\_p\cong H\_{ét}^1(\mathcal{O}\_F,\mathbb{Z}\_p(n))
$$
by the Bloch-Kato conjecture, as proven by Rost, Suslin, Voevodsky and Weibel. So my question becomes:



> 
> when is
>  $$
> H\_{ét}^1(\mathcal{O}\_F,\mathbb{Z}\_p(n))^G \cong H\_{ét}^1(\mathcal{O}\_K,\mathbb{Z}\_p(n))?
> $$
> 
> 
> 


A concise reference to a place that summarises everything we know about this would be great! Or maybe it is clear to the experts that this is always true? That would be even better.



Galois descent for K-groups (or for étale cohomology groups)
Sorry to answer my own question, but according to [these notes](http://www.ictp.it/~pub_off/lectures/lns015/Kolster/Kolster_Final.pdf) by Manfred Kolster, Proposition 2.9, the étale cohomology groups in fact always satisfy Galois descent. Should have googled harder before asking.


