Rationality problem for norm one tori of tensor products of \'etale algebras and Hasse norm principle

TL;DR

This paper investigates the rationality properties of norm one tori associated with tensor products of étale algebras, establishing conditions under which their rationality is preserved and applying results to the Hasse norm principle over global fields.

Contribution

It proves that under certain gcd conditions, the stable and retract rationality of individual tori implies the same for their tensor product and associated norm one tori, with applications to the Hasse norm principle.

Findings

Stable and retract rationality are preserved under tensor products when gcd conditions are met.

The Hasse norm principle holds for tensor products of étale algebras over global fields.

Explicit criteria are provided for the rationality of norm one tori in this context.

Abstract

Let $k$ be a field. Let $A=\prod_{i=1}^r K_i$ and $B=\prod_{j=1}^s E_j$ be \'etale $k$-algebras where $K_i$ and $E_j$ are finite separable field extensions of $k$ with $[K_i:k]=m_i$ and $[E_j:k]=n_j$. Let $\mathcal{T}_A=R^{(1)}_{A/k}(\mathbb{G}_m)$ be the norm one torus of the \'etale $k$-algebra $A$. We prove that if $\gcd(m_i,n_j\mid 1\leq i\leq r, 1\leq j\leq s)=1$ and $\mathcal{T}_A$ and $\mathcal{T}_B$ are stably $($resp. retract$)$ $k$-rational, then the algebraic $k$-torus $\mathcal{T}_A\otimes \mathcal{T}_B$ and the norm one torus $\mathcal{T}_{A\otimes B}$ are stably $($resp. retract$)$ $k$-rational. We then give detailed applications to the case of norm one tori of field extensions. In particular, if $k$ is a global field, then the Hasse norm principle holds for $(A\otimes B)/k$.