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 RK/k(1)(Gm) of K/k whose Galois closures L/k are dihedral extensions with the aid of Endo and Miyata [17, Theorem 1.5, Theorem 2.3] and Endo [15, 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.