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