Let $K$ be a number field of degree $d=[K,\mathbb{Q}]$ and suppose that the class number $h_K$ of $K$ is comprime with $d$. Let $p$ a prime of $\mathbb{Q}$ and assume that $(p)=\mathcal{P}^d$ is totally ramified in $K$, where $\mathcal{P}$ is the unique prime of $K$ above $p$. Show that $[\mathcal{P}]=1$ in the class group $Cl_K$ of $K$.
Since $\mathcal{P}^d$ is totally ramified, then $\mathcal{O}_K / \mathcal{P}^d \cong \mathbb{Z}/p\mathbb{Z}$ and so It is a maximal ideal.
I think that I have to prove that $\mathcal{P}=p\mathcal{O}_K$ but I have no idea how to use $(p,h_K)=1$.
some ideas?