Inertia and Decomposition Field in $\mathbb{Q}(\omega=exp(2\pi i/m)$, where $m=p^{k}n,(p,n)=1$.

let $ \omega=exp(2\pi i/m),m=p^{k}n,(p,n)=1$ . We know that the Galois group $ \mathbb{Q}(\omega)$ over $ \mathbb{Q}$ is isomorphic to $ (\mathbb{Z}_{m})^\times$ , which is naturally isomorphic to $ \mathbb{Z}_{p^k}^{\times}\times(\mathbb{Z}_{n})^{\times}$ . My question is: how does one describe the decomposition group, $ D$ and inertia group $ E$ with respect to $ p$ in terms of $ (\mathbb{Z}_{p^k})^\times$ and $ (\mathbb{Z}_{n})^\times$ ?

Here is my attempt: Let $ Q$ be any prime lying over $ p$ in $ \mathbb{Q}(\omega)$ . $ p$ splits completely in $ \mathbb{Q}(\omega^{p^k})$ . Hence, $ \mathbb{Q}(\omega^{p^k})$ lies in the fixed field of the decomposition group $ L_{D}$ , which is a subfield of $ L_{E}$ (the fixed field of the inertia group). Therefore, we have $ [L_{E}:\mathbb{Q}]\geq [\mathbb{Q}(\omega^{p^{k}}):\mathbb{Q}]=\phi(n)$ . But we know that $ [L:L_{E}]=\phi(p^{k})$ , which implies $ L_{E}=\mathbb{Q}(\omega^{p^{k}})$ . Hence, $ E$ is isomorphic to $ \mathbb{Z}_{p^k}$ (need to use Galois theory).

The part that i am having trouble with is how does one describe $ D$ ? I know that $ D/E$ will be a subgroup of $ \mathbb{Z}_{n}^\times$ , and also cyclic, but i am not sure how to describe it. Thanks.