FC--Is there some source you recommend to read up on this? I would be happy to learn more about the totally real case, as well as these finitely many examples. In particular, I would like to be able to have a class of such fields. It would be wonderful if [K:Q] > or = 3. Is it true that there are only finitely many fields that are not totally real, or only finitely many fields that are totally real with [K:Q] >2?

Sorry about the "unknown control sequence". I mean in the cyclotomic extension of $\mathbb{Q}$ one gets by adding the f(K)th roots of unity.

With respect to the last comment, are you trying to say that if K has conductor $f(K)$, then the genus class field is contained in $\Q(\zeta_f(K))?$