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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?

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Are you interested in the full kernel of Norm or only on the kernel on the $p$-Sylow of the corresponding groups (I.e. $\otimes\mathbb{Z}_p$ of everything)? And by "compute" do you mean more or less algorithmically or rather theoretically, like giving upper/lower bounds of these relative groups building upon some other invariants of $K$? –  Filippo Alberto Edoardo Nov 5 '12 at 14:46
    
a) The p-part would be fine; b) I was thinking of an algorithm. –  Franz Lemmermeyer Nov 5 '12 at 17:38

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