## 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.