## 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.

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## Does Levitate cancels inertia?

According to the spell description:

Levitate allows you to move yourself, another creature, or an object up and down as you wish. A creature must be willing to be levitated, and an object must be unattended or possessed by a willing creature. You can mentally direct the recipient to move up or down as much as 20 feet each round; doing so is a move action. You cannot move the recipient horizontally, but the recipient could clamber along the face of a cliff, for example, or push against a ceiling to move laterally (generally at half its base land speed).

Source: https://www.d20pfsrd.com/magic/all-spells/l/levitate/

My understanding is that while the description says you can’t make something/someone/oneself move horizontally through this spell, there’s nothing stopping the object from moving horizontally by its own device.

So, imagine the following case:

A drow runs at her maximum speed, and jumps over a cliff, activating her Levitate spell-like ability.

1. Will the character’s inertia make her move horizontally, while she keep her altitude constant (or rising, or going down) by carefully using Levitate?
2. If there’s no obstacle, will she even stop?
3. How does that compare to Fly? (https://www.d20pfsrd.com/magic/all-spells/f/fly/)