The rationality problem for multinorm one tori, II

TL;DR

This paper studies the stable and retract rationality of multinorm one tori linked to finite étale algebras, providing criteria based on degrees and Galois group structures.

Contribution

It offers new criteria for rationality properties of multinorm one tori, extending classical results to broader Galois extension cases.

Findings

Tori are stably rational when gcd of degrees is 1.

Provides sufficient conditions for failure of retract rationality when gcd > 1.

Generalizes classical results to Galois groups with cyclic Sylow subgroups and dihedral groups.

Abstract

We investigate the stable and retract rationality of multinorm one tori associated to finite {\'e}tale algebras. Our results are organized according to the greatest common divisor $d$ of the degrees of the factors. We show that these tori are stably rational for $d=1$, and obtain a criterion for retract rationality that can be attributed to our previous results. For $d>1$, we provide sufficient conditions for the failure of retract rationality. We further generalize results of Endo--Miyata (1975) and Endo (2011) by giving an equivalent condition for multinorm one tori to be stably rational under the assumption that they split over Galois extensions with Galois groups in which all Sylow subgroups are cyclic. A similar result also holds when they split over dihedral Galois extensions.