In the explicit construction of Hilbert $p$-class fields of a number field $K$ it is not so much the class group of $K$ or that of $L = K(\zeta_p)$ that is needed but the relative class group of the extension $L/K$, that is, ideal classes from $L$ whose norm down to $K$ is trivial. Currently it seems the best we can do is compute the full class group of $L$ and then look at the pieces that we need. I have often wondered whether there is a better way of computing the relative class group (the part killed by the norm) of an extension, or whether there are at least ideas of how to do it.

In a similar vein, is there a good way of computing relative units (those whose norm down to $K$ is a root of unity) without computing the full unit group?