I 've seen the following statement in
https://mathoverflow.net/questions/94480/generalization-of-hilbert-94-and-capitulation and https://mathoverflow.net/questions/63465/where-does-the-principal-ideal-theorem-from-cft-go .
"There is a generalization of the principal ideal theorem to ray class groups:
Let $K$ be a number field, $\frak{m}$ a modulus for $K$, $L:=K(\frak m)$ the ray class field modulo $\frak m$, and $\frak n$ the image of $\frak m$ in $L$. Then any element in the ray class group modulo $\frak m$ of $K$ becomes trivial in the ray class group modulo $\frak n$ of $L$. The proof is almost identical to that of Principal ideal theorem, reducing to the same group theoretic statement."
How can I prove this? In my thinking, it is required that $Gal(K({\frak{m}})/ K )=Gal(L({\frak{n}})/K)^{ab}$ where $L(\frak n)$ is the ray class field modulo $\frak n$. But why does this hold?