Constructing abelian extensions with prescribed norms

Given a number field $K$, finite abelian group $G$ and finitely many elements $\alpha_1,\ldots,\alpha_t\in K$, we construct extensions $L/K$ with Galois that realise all of the $\alpha_1,\ldots,\alpha_t$ as norms in $L$. In particular, this shows existence such for any given parameters. Our approach relies on class theory recent formulation Tate's characterisation Hasse norm principle, local-global principle norms. The constructions are sufficiently explicit to be implemented computer, illustrate them concrete examples.