Let $k$ be a number field and $p$ be a rational prime. Then consider the extension $k(\mu_{p^m}) \mid k$ of adjoining all roots of unity of degree $p^m$ to $k$. Assuming that $\mu_{p^m} \cap k$ is non-trivial:
How to find the Galois group of this extension? Also which primes of $k$ can ramify/decompose in this extension? I don't know how to tackle this problem properly, but it should not be that difficult.
I know so far that the extension $k(\mu_{p^m}) \mid k$ is of degree less than $\varphi(p^m) = p^{m-1}(p-1)$, where $\varphi$ denotes the Eulerian Phi-function.
Is the Galois group now isomorphic to $\mathbb{Z} / (p^{m-1}(p-1)) \mathbb{Z}$, since the extension by roots of unity must be cyclic?
Thanks, Tom