How to show that $G=\mathrm{Gal}[\Bbb Q(\xi_{p^n})/\Bbb Q(\xi_{p^m})]$ cyclic? Here $p \not = 2$ is a prime number and $n>m>0$ are natural numbers. $\xi_{p^n}$ and $\xi_{p^m}$ are cyclotomic roots.
I know that the order of $ G $ is $|G|=p^{n-m}$
How I found $a \in G$ such that $a^{p^{n-m}}=1$ and how generally I found such $a$ in other cyclic cases?
The group $G$ is a subgroup of $\mathrm{Gal}(\Bbb Q(\zeta_{p^n}) / \Bbb Q)$, which is isomorphic to $(\Bbb Z/p^n \Bbb Z)^{\times}$ (see this related question). This is a cyclic group when $p>2$ is prime. Since any subgroup of a cyclic group is cyclic, we are done.