Hasse norm principle for Galois dihedral extensions

TL;DR

This paper investigates the Hasse norm principle and weak approximation properties for dihedral Galois extensions of number fields, providing a comprehensive description of these phenomena in this specific algebraic setting.

Contribution

It offers a general description of the Hasse norm principle and weak approximation for dihedral Galois extensions, extending understanding in this area.

Findings

Characterization of the Hasse norm principle for dihedral extensions

Conditions under which weak approximation holds for the associated norm one torus

New insights into the arithmetic of dihedral Galois extensions

Abstract

Let $L/k$ an Galois extension of number fields with Galois group isomorphic to a dihedral group of order $2n$. In this note, we give a general description of the Hasse norm principle for $L/k$ and the weak approximation for the norm one torus $R^1_{L/k}(\mathbb{G}_m)$ associated to $L/k$.