We consider the problem of whether the norm one torus defined by a finite separable field extension is stably (or retract) rational over . This has already been solved for the case where is a Galois extension. In this paper, we solve the problem for the case where is a non-Galois extension such that the Galois group of the Galois closure of is nilpotent or metacyclic.
"The rationality problem for norm one tori." Nagoya Math. J. 202 83 - 106, June 2011. https://doi.org/10.1215/00277630-1260459