Let $K/k$ and $L/k$ be abelian extensions of number fields. Prove that they have the same ramification iff $$N_{K/k}P_K=N_{L/k}P_L$$ as subgroups of $I_k$.
Where $I_k$ is the group of all ideals relatively prime to the discriminant of $K/k$ and $P_K$ all principal ideals in $K$, relatively prime to the discriminant of $K/k$.
Seasons greetings.