Rationality problem for norm one tori for dihedral extensions

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