Suppose $K$ is a number field. Let $q$ be a prime of $K$. Let $ K [q] $ be the ray class field of $K$ of modulus $q$. Someone claims that $K[q]$ is totally ramified at $q$ over the hilbert class field of $K$.
Moreover, in Gross and Zagier's paper about Kolyvagin's work on modular elliptic curves, there is a similar result. Suppose $l$ is a prime. Then the ring class field of an imaginary quadratic field $K$ of modulus $l$ is totally ramified at $l$ over the hilbert class field of $K$.
Both results are claimed that they can be solved by class field theory. However I don't know how to see it.
I assume that by totally ramified at $q$ you mean that $K[q]/H$, where $H$ is the Hilbert class field of $K$, is ramified only at primes above $q$ and that no intermediate extension $K[q] \supset K' \supset H$ is unramified.
This follows from the description of $H$ as the maximal abelian everywhere unramified extension of $K$, and that $K[q]/K$ is the maximal abelian extension ramified only at primes above $q$. It follows that $K'$ is abelian over $K$ as well. If $K'/K$ was unramified, then this would contradict the maximality of $H$!
The same goes for the ring class field, which can be described as a subfield of the ray class field of the same modulus.