I am studying algebraic number theory/class field theory. There I need the following result
Let $q$ be a prime in $K$, a number field, and $F$ a finite abelian extension of $K$ such that $F$ contains the Hilbert Class of $K$ and $q$ splits completely in $F$. Let $M$ be a finite extension of $F$, abelian over $K$, such that in $M/F$ all primes above $q$ are totally, tamely ramified and no other primes ramify. Then, $Gal(M/F)$ is cyclic.
I have no idea how to prove it. I tried to use information from cyclotomic fields, because $\mathbb{Q}(\zeta_{p^{n}})/\mathbb{Q}$ is cyclic and ramifies only in $p$. However, I am not sure how to use it and I don't if $p$ totally ramifies.