Let $\zeta \in \overline{\mathbb{Q}}$ be a primitive 8th root of unity and let $K = \mathbb{Q}(\zeta)$. Let $N\_{K/\mathbb{Q}}$ be the corresponding norm.


Consider the three-dimensional $\mathbb{Q}$-torus defined by $N\_{K/\mathbb{Q}}(x) = 1$ for $x \in K^\times$. Is it rational over $\mathbb{Q}$?


I think the answer is no, but I don't have a proof.



Rationality of three-dimensional torus
The torus is not rational. 


This is due to B.E. Kunyavskii, *On Tori with a biquadratic splitting field,* Izv. Akad. Nauk SSSR Ser. Mat 42 (1978), 580-587; English translation Math USSR-IZV. 12(1978), 536-542.


An easier to find reference is V.E. Voskresenskii's book *Algebraic Groups and their Birational Invariants.* See Section 4.8, concluding with Example 4.8.1, at the top of page 54. Also, see the beginning of Section 4.10. 


The argument given in that book actually shows something more general: the Norm 1 torus is non-rational, if the Galois group of your extension is $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$, for any prime $p$. This is what is done in Section 4.8. 


In Section 4.10, Voskresenskii recalls the work of Kunyavskii, where it is shown that the Norm 1 torus for a biquadratic extension is not stably rational.


