Special case of Kronecker–Weber theorem.

Question

Let $K$ be a number field contained in $m^{th}$ cyclotomic field, that is $K \subset \Bbb{Q}(\omega)$ where $\omega$ is a primitive $m^{th}$ root of unity. Let $p^k$ be the exact power of a prime $p$ dividing $m$, and suppose that $p$ is unramified in $K$. Then how to show that $K$ is contained in $(\frac{m}{p^k})^{th}$ cyclotomic field.

Answer 1

Hint: Think about what subgroup of $(\mathbb{Z}/m\mathbb{Z})^\times$ the field $K$ must correspond to.