Rationality problem for norm one tori for dihedral extensions

TL;DR

This paper completely resolves the rationality problem for norm one tori associated with dihedral Galois extensions, using techniques from Endo and Miyata, and refines previous results on stable rationality.

Contribution

It provides a complete classification of the rationality of norm one tori for dihedral extensions and refines existing proofs of stable rationality cases.

Findings

Complete solution to the rationality problem for dihedral extensions.

Refined proofs of stable rationality cases.

Application of Endo and Miyata's techniques to this problem.

Abstract

We give a complete answer to the rationality problem (up to stable $k$-equivalence) for norm one tori $R^{(1)}_{K/k}(\mathbb{G}_m)$ of $K/k$ whose Galois closures $L/k$ are dihedral extensions with the aid of Endo and Miyata [EM75, Theorem 1.5, Theorem 2.3] and Endo [End11, Theorem 2.1]. By using a similar technique, we give refinements of the proof of stably rational cases of Endo and Miyata's theorems as an appendix of the paper.