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.