I've been trying to figure out the following problem. Given a prime $p$, and $\zeta_m = e^{2\pi i /m}$, and the factorization $m = p^kn$ with $(p,n) = 1$, then we know that the Galois group of $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is isomorphic to the group of units $(\mathbb{Z}/m\mathbb{Z})^{\times}$ which is in turn isomorphic to $(\mathbb{Z}/p^k\mathbb{Z})^{\times} \times (\mathbb{Z}/n\mathbb{Z})^{\times}$. I'm trying to see what the decomposition group $D$ and the inertia group $E$ both over $p$ look like when we consider the Galois group in terms of $(\mathbb{Z}/p^k\mathbb{Z})^{\times} \times (\mathbb{Z}/n\mathbb{Z})^{\times}$ instead of in terms of $(\mathbb{Z}/m\mathbb{Z})^{\times}$.
In the case where $k=0$, I think $E$ must be the trivial group since $p$ is unramified. I think the decomposition group $D$ would be the group of order $\phi(m)/f$ in $(\mathbb{Z}/m\mathbb{Z})^{\times}$ where $f$ is the order of $p \mod \phi(m)$. But I'm not sure where to go from here. Any tips or hints on where to proceed would be greatly appreciated.