How to show that $G=\mathrm{Gal}[\Bbb Q(\xi_{2^n})/\Bbb Q(\xi_{2^m})]$ cyclic?

How to show that $G=\mathrm{Gal}[\Bbb Q(\xi_{2^n})/\Bbb Q(\xi_{2^m})]$ cyclic? $n>m>1$ are natural numbers. $\xi_{2^n}$ and $\xi_{2^m}$ are cyclotomic roots.

I know that the order of $G$ is $|G|=2^{n-m}$

• The only thing I succeed at it is to show that $G \cong Z_2 \times Z_{2^{n-2}}/Z_2 \times Z_{2^{m-2}}$ but why this is cyclic? – Daniel Vainshtein Jun 21 '18 at 16:36
• The Galois groups are generated by the automorphisms $\xi \mapsto \xi^{-1}$ and $\xi \mapsto \xi^{5}$. – sharding4 Jun 21 '18 at 16:58