Let $K$ be a number field, $v$ a finite place. Let $\bar{v}$ be the unique extension of $v$ from $K_v$ to $\overline{K_v}$. We fix an embedding $i:\overline{K} \to \overline{K_v}$, then we have a valuation $v^{\bullet}$ on $\overline{K}$ which is the restriction of $\bar{v}$ to $\overline{K}$.
Define the decomposition group $G_v:=\{\sigma \in \mathrm{Gal}(\overline{K}/K) | \sigma (v^{\bullet})=v^{\bullet}\}$
Now my question is how to show the decomposition group $G_v$ of $v$ is isomorphic to $\mathrm{Gal}(\overline{K_v}/K_v)$ ?
Thanks for any help.