Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$? I am going to be on the road soon, so pleas don't be offended if I don't respond quickly to a comment.



Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$?
The genus class field of an extension $K/F$ is defined to be the largest extension $L/K$ with the following properties:


1. $L/K$ is unramified
2. $L$ is the compositum of $K/F$ and an abelian extension $A/F$.


Thus the quick answer to your question is: the Hilbert class field of $K$ is abelian over ${\mathbb Q}$ if and only if the Hilbert class field of $K$ coincides with its genus class field. 


The not-so-quick answer would tell you more about the construction of the genus class field. For abelian extensions of the rationals, the construction is easy: everything you'd like to know should be contained in Frölich's book


* *Central extensions, Galois groups, and ideal class groups of number fields* 
AMS 1983


Basically you will have to look for the largest abelian extension of ${\mathbb Q}$ with the same conductor as $K/{\mathbb Q}$.


