Comments

Wafa Wei's results on Hilbert's 12th problem are available in her preprint.

Erratum

There is an arrow missing in the diagram of Lemma 11.1 on page 24.

p.9: The last part of Proposition 3.5 should read: moreover, $e_{\Phi}(\tau)$ depends only on the effect of $\tau$ on $E$ and is $1$ if $\tau|E=\mathrm{id}$ (i.e., replace $K_{0}$ with $E$, the reflex field).