Let $K$ be a finite extension of $\mathbb{Q}\_p$.


Is the centraliser of $\operatorname{Gal}(\overline{K}/K)$ in $\operatorname{Gal}(\overline{\mathbb{Q}\_p} / \mathbb{Q}\_p)$ trivial ?
If yes, how can I show it ?



Centraliser of an absolute Galois group
For an extension $L/\mathbf{Q}\_p$, let $G\_L$ denote the absolute Galois group $\mathrm{Gal}(\overline{L}/L)$.


If $\sigma \in G\_{\mathbf{Q}\_p}$ acts centrally on $G\_K$, then it also acts centrally on the subgroup $G\_L \subset G\_K$ for any finite $L/K$. But then it also acts trivially on the abelianization of $G\_L$. By class field theory, the abelianization of $G\_L$ may be identified with the pro-finite completion of $L^{\times}$, and moreover this isomorphism is compatible with the action of $G\_{\mathbf{Q}\_p}$. Hence $\sigma$ must also act trivially on $L^{\times}$ for any finite $L/\mathbf{Q}\_p$, and thus $\sigma$ is trivial.


