Constructing abelian extensions with prescribed norms

Abstract

Given a number field K K , a finite abelian group G G and finitely many elements α 1 , … , α t ∈ K \alpha _1,\ldots ,\alpha _t\in K , we construct abelian extensions L / K L/K with Galois group G G that realise all of the elements α 1 , … , α t \alpha _1,\ldots ,\alpha _t as norms of elements in L 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.