Let $G_q$ be the decompostion subgroup $\text{Gal}(\overline{\mathbb{Q}}_q/\mathbb{Q}_q)$ where $q$ be a prime. Let $p$ be a prime and $p \ne q$ and we take $\mu_p$ be the group $p$-th roots of unity in $\overline{\mathbb{Q}}_q$. Clearly there's a natural action of $G_q$ on $\mu_p$.
My aim is to show:
Action of $G_q$ on $\mu_p$ is unramified.
My idea:
Let $\zeta_p$ be a primitive $p$-th root of unity in $\overline{\mathbb{Q}}_q$. We consider the extention $\mathbb{Q}_q(\zeta_p)/\mathbb{Q}_q$. This is clearly a Galois extention and of degree $p-1$. After reduction we get the extention $\mathbb{F}_q(\zeta_p)/\mathbb{F}_q$ and which is also of degree $p-1$ as $\text{gcd}(p, q) =1$. So, the extension $\mathbb{Q}_q(\zeta_p)/\mathbb{Q}_q$ is unramified. Hence the action of $G_q$ is unramified on $\mu_p$.
Does this argument works? Are there better ways to do this? I'm a just a beginner in this area. Any hint or idea will be very helpful. Thanks in advance.