$p$-retract Rationality and Norm One Tori

We study whether the norm one torus associated with a finite separable non-Galois field extension $K\mathrm{/}k$ is $p$-retract rational over $k$ for a prime $p$, focusing on the case where the Galois group of the Galois closure of $K\mathrm{/}k$ is either the symmetric or the alternating group.