Rationality problem for norm one tori of tensor products of \'etale algebras and Hasse norm principle
Pith reviewed 2026-05-19 22:46 UTC · model grok-4.3
The pith
When degrees of two étale algebras over k are coprime, stable or retract rationality of their norm one tori passes to the tensor product torus and the norm one torus of the tensor product algebra.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Let A and B be étale k-algebras whose component field degrees satisfy gcd of all m_i and n_j equal to one. If the norm one torus of A and the norm one torus of B are stably k-rational, then both the tensor product torus and the norm one torus of A tensor B are stably k-rational; the same statement holds with stably replaced by retract. In particular, when k is a global field the Hasse norm principle holds for the extension given by A tensor B.
What carries the argument
The norm one torus T_{A/k} defined as the kernel of the norm map from the Weil restriction of the multiplicative group from the étale algebra A to k; tensor product of algebras induces a corresponding operation on these tori that preserves the rationality properties under the coprimeness hypothesis.
If this is right
- Detailed applications exist for norm one tori attached to single field extensions rather than products.
- The Hasse norm principle holds for A tensor B over any global field k whenever the degree condition and rationality hypotheses are met.
- The same preservation holds when stable rationality is weakened to retract rationality.
- The result applies directly to products of cyclic extensions whose degrees are pairwise coprime.
Where Pith is reading between the lines
- One could build new families of rational norm one tori by iteratively tensoring known rational examples with extensions of coprime degree.
- The coprimeness hypothesis likely removes a common prime factor in the Galois module that would otherwise produce a nontrivial Brauer-Manin obstruction or non-rationality invariant.
- Explicit low-degree checks over Q, such as tensoring a quadratic extension with a cubic extension, could verify the result by direct computation of the torus equations.
Load-bearing premise
The coprimeness condition that the greatest common divisor of all the extension degrees appearing in A and in B equals one must hold for the rationality to pass to the tensor product.
What would settle it
An explicit pair of étale algebras over a number field whose degrees are coprime, with both norm one tori stably rational, yet the norm one torus of their tensor product failing to be stably rational, would refute the claim.
read the original 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. In particular, if $k$ is a global field, then the Hasse norm principle holds for $(A\otimes B)/k$. We then give detailed applications to the case of norm one tori of field extensions. We investigate more general situations $T_1\otimes T_2$ for algebraic $k$-tori $T_1$ and $T_2$ by introducing a useful invariant of a $G$-lattice: its permutation order.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves that if A = ∏ K_i and B = ∏ E_j are étale k-algebras with [K_i : k] = m_i and [E_j : k] = n_j satisfying gcd(m_i, n_j for all i,j) = 1, and if the norm-one tori T_A and T_B are stably (resp. retract) k-rational, then both the tensor-product torus T_A ⊗ T_B and the norm-one torus T_{A⊗B} are stably (resp. retract) k-rational. The result is applied to field extensions and, when k is global, yields the Hasse norm principle for (A ⊗ B)/k via the known equivalence between retract rationality of the norm-one torus and vanishing of the relevant Sha group.
Significance. The coprimeness hypothesis permits an explicit transfer of stable/retract rationality via Galois-cohomological and torus-isogeny constructions, extending existing rationality criteria to composite étale algebras. The self-contained argument and the direct implication for the Hasse norm principle on global fields constitute a concrete advance in the study of algebraic tori.
minor comments (2)
- [Introduction] §1: the definition of the tensor product torus T_A ⊗ T_B is introduced without an explicit reference to the underlying Galois module construction; a one-sentence reminder would aid readability.
- [Applications] §4, after the statement of the main theorem: the reduction to the case of field extensions is sketched but the precise identification of the Galois action on the character lattice of T_{A⊗B} is not written out; adding the lattice description would make the application section self-contained.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of our manuscript, including the summary of the main results on the preservation of stable and retract rationality for norm-one tori under tensor products when the coprimeness condition holds, and the direct application to the Hasse norm principle over global fields. The recommendation for minor revision is noted, and we will incorporate any editorial or minor improvements in the revised version. As the report contains no major comments, we have no specific points to address point-by-point.
Circularity Check
No significant circularity detected in derivation chain
full rationale
The paper establishes its main theorem via explicit Galois-cohomological constructions and torus-isogeny arguments that transfer stable or retract rationality from T_A and T_B to T_A ⊗ T_B and T_{A⊗B} under the coprimeness hypothesis. This chain does not reduce any claimed prediction or result to a fitted parameter, self-definition, or load-bearing self-citation; the Hasse norm principle application follows from an external known equivalence between retract rationality and vanishing of the Sha group. The argument is self-contained against external benchmarks with no reduction of outputs to inputs by construction.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math A and B are étale k-algebras, i.e., finite products of finite separable field extensions of k.
- standard math Stable rationality and retract rationality are the usual birational notions for algebraic varieties over k.
discussion (0)
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