Constructing abelian extensions with prescribed norms

Given a number field K, a finite abelian group G and finitely many elements α1,…,αt ∈ K, we construct abelian extensions L/K with Galois group G that realise all of the elements α1,…,αt as norms of elements in L. In particular, this shows existence of such extensions for any given parameters. Our approach relies on class field theory and a recent formulation of Tate’s characterisation of the Hasse norm principle, a local-global principle for norms. The constructions are sufficiently explicit to be implemented on a computer, and we illustrate them with concrete examples