Rationality problem for norm one tori for dihedral extensions

Abstract

We give a complete answer to the rationality problem (up to stable $k$-equivalence) for norm one tori $R_{K/k}^{(1)}({\mathbb{G}}_m)$ 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.

arXiv comment11 pages, typos corrected, to appear in J. Algebra. arXiv admin note: text overlap with arXiv:1811.01676, arXiv:1910.01469, arXiv:1811.02145, arXiv:1210.4525, arXiv:2112.02280