Constructing abelian extensions with prescribed norms

TL;DR

This paper develops a method using class field theory and Tate's Hasse norm principle to explicitly construct abelian extensions of number fields with prescribed norm properties, demonstrating their existence and providing computational examples.

Contribution

It introduces an explicit construction technique for abelian extensions with specified norm conditions, expanding the toolkit for number field extension problems.

Findings

Constructed abelian extensions realizing given elements as norms.

Proved existence of such extensions for arbitrary parameters.

Provided explicit, implementable algorithms with concrete examples.

Abstract

Given a number field $K$, a finite abelian group $G$ and finitely many elements $\alpha_1,\ldots,\alpha_t\in K$, we construct abelian extensions $L/K$ with Galois group $G$ that realise all of the elements $\alpha_1,\ldots,\alpha_t$ 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.