The rationality problem for norm one tori

Abstract

We consider the problem of whether the norm one torus defined by a finite separable field extension $K/k$ is stably (or retract) rational over $k$. This has already been solved for the case where $K/k$ is a Galois extension. In this paper, we solve the problem for the case where $K/k$ is a non-Galois extension such that the Galois group of the Galois closure of $K/k$ is nilpotent or metacyclic.