Let $K$ be a number field, $\alpha _1,...,\alpha _t \in K$ and $G$ a finite abelian group. We explain how to construct explicitly a normal extension $L$ of $K$ with Galois group $G$, such that all of the elements $\alpha_{i}$ are norms of elements of $L$. The construction is based on class field theory and a recent formulation of Tate's criterion for the validity of the Hasse norm principle. This is joint work with Rodolphe Richard (UCL).