Let $K$ be an abelian extension of $\mathbb{Q}$ with $[K:\mathbb{Q}] = p^m$. Suppose $q$ is a prime $\neq p$ which is ramified in $K$. Let $Q$ be a prime of $K$ lying over $q$.
Prove that $e(Q|q)$ divides $q-1$ and that the $q$th cyclotomic field has a unique subfield $L$ of degree $e(Q|q)$ over $\mathbb{Q}$.
Thanks in advance
Since $K/\mathbb{Q}$ is galois, then $e=e(Q/q)$ must divide $[K:\mathbb{Q}](=efg).$ In the other hand by Kronecker-Weber theorem ther is some integre $m$ such that $K\subset\mathbb{Q}(\zeta_m)$ if $q^{\alpha}|| m$ we have $$q\mathcal{O}_K=q\mathbb{Z}[\zeta_m]=(1-\zeta_m)^{\varphi(q^{\alpha})}\;\;\text{and so}\;\;e(\mathbb{Q}(\zeta_m)/q)=\varphi(q^\alpha)=q^{\alpha-1}(q-1)\;\; \text{where $\varphi$ is Euler's function}$$ Therefore $e|\varphi(q^{\alpha})$ this implies : $e|(\gcd(\varphi(q^{\alpha}),p^m))$ since $p\neq q$ we conclude that $e|(q-1).$
For the second question: $G=\mathrm{Gal}(\mathbb{Q}(\zeta_q)/\mathbb{Q})\cong (\mathbb{Z}/q\mathbb{Z})^*\cong \mathbb{Z}/(q-1)\mathbb{Z}$ is a cyclic group since $q-1=k.e$ then $\mathbb{Z}/(q-1)\mathbb{Z}$ admit a unique subgroup $H$ of order $k,$ and takes $L=\mathbb{Q}(\zeta_q)^{H}.$