Number fields with prescribed norms (with an appendix by Yonatan Harpaz and Olivier Wittenberg)

We study the distribution of extensions a number field $k$ with fixed abelian Galois group $G$, from which given finite set elements are norms. In particular, we show existence such extensions. Along way, that Hasse norm principle holds for $100\%$ $G$-extensions $k$, when ordered by conductor. The appendix contains an alternative purely geometric proof our result.