pith. sign in

arxiv: 2605.17427 · v1 · pith:LKKZ5TJBnew · submitted 2026-05-17 · 🧮 math.AG · math.NT

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

Mathieu Florence , Akinari Hoshi , Aiichi Yamasaki This is my paper

Pith reviewed 2026-05-19 22:46 UTC · model grok-4.3

classification 🧮 math.AG math.NT
keywords norm one toristable rationalityretract rationalityHasse norm principleétale algebrastensor productsalgebraic torirationality problems
0
0 comments X

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.

The paper proves a preservation theorem for rationality of algebraic tori attached to étale algebras. If the greatest common divisor of all the degrees of the finite separable extensions making up two such algebras A and B equals one, and if the norm one tori of A and of B are stably rational or retract rational, then both the product torus and the norm one torus of A tensor B inherit the same rationality property over k. When k is a global field this also yields the Hasse norm principle for the tensor product algebra. A reader would care because these rationality statements control whether certain tori have dense rational points and whether Galois cohomology obstructions vanish, which in turn governs arithmetic questions about norms and extensions.

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

These are editorial extensions of the paper, not claims the author makes directly.

  • 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. 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 $(A\otimes B)/k$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

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)
  1. [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.
  2. [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

0 responses · 0 unresolved

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

0 steps flagged

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

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background from algebraic geometry without introducing new free parameters or invented entities. The central claim rests on definitions of étale algebras and rationality notions that are taken from prior literature.

axioms (2)
  • standard math A and B are étale k-algebras, i.e., finite products of finite separable field extensions of k.
    Standard setup for norm one tori in the theory of algebraic groups over fields.
  • standard math Stable rationality and retract rationality are the usual birational notions for algebraic varieties over k.
    Background definitions from birational geometry invoked in the statement.

pith-pipeline@v0.9.0 · 5775 in / 1529 out tokens · 50789 ms · 2026-05-19T22:46:23.779974+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

  1. [1]

    [Arn84] J. E. Arnold, Jr, Groups of permutation projective dimension two , Proc. Amer. Math. Soc. 91 (1984) 505–509. [Azu50] G. Azumaya, Corrections and supplementaries to my paper concerning Kru ll-Remak-Schmidt’s theorem, Nagoya Math. J. 1 (1950) 117–124. [Bar81a] H.-J. Bartels, Zur Arithmetik von Konjugationsklassen in algebraischen G ruppen, J. Algebr...

    work page 1984

  2. [2]

    Beauville, The L¨ uroth problem, Rationality problems in algebraic geometry, 1–27, Lectur e Notes in Math., 2172, Fond

    [Bea16] A. Beauville, The L¨ uroth problem, Rationality problems in algebraic geometry, 1–27, Lectur e Notes in Math., 2172, Fond. CIME/CIME Found. Subser., Springer, Cham, 2016 . [Ben98] E. Beneish, Induction theorems on the stable rationality of the center o f the ring of generic matrices , Trans. Amer. Math. Soc. 350 (1998) 3571–3585. [BLB91] C. Bessen...

    work page 2016

  3. [3]

    [Bou90] N

    v ii+422 pp. [Bou90] N. Bourbaki, Algebra II, Chapters 4–7 , Translated from the French by P. M. Cohn and J. Howie, Elem. M ath. (Berlin), Springer-Verlag, Berlin, 1990, vii+461 pp. NORM ONE TORI OF TENSOR PRODUCTS OF ´ETALE ALGEBRAS AND HASSE NORM PRINCIPLE 35 [Bro82] K. S. Brown, Cohomology of groups , Graduate Texts in Mathematics, 87, Springer-Verlag,...

    work page 1990

  4. [4]

    Colliot-Th´ el` ene, A

    [CTS21] J.-L. Colliot-Th´ el` ene, A. N. Skorobogatov, The Brauer-Grothendieck group , Ergeb. Math. Grenzgeb. (3) 71, Springer, Cham, 2021, xv+453 pp. [CK00] A. Cortella, B. Kunyavskii, Rationality problem for generic tori in simple groups , J. Algebra 225 (2000) 771–793. [CR81] C. W. Curtis, I. Reiner, Methods of representation theory, vol. I, with appli...

    work page 2021

  5. [5]

    Demarche, D

    [DW14] C. Demarche, D. W ei, Hasse principle and weak approximation for multinorm equat ions, Israel J. Math. 202 (2014) 275–293. [DP87] Yu. A. Drakokhrust, V. P. Platonov, The Hasse norm principle for algebraic number fields , (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986) 946–968; translation in Math. USSR-Izv. 29 (1987) 299–322. [End01] S. Endo, On ...

    work page 2014

  6. [6]

    [Ger77] F

    (http://www.gap-system.org). [Ger77] F. Gerth III, The Hasse norm principle in metacyclic extensions of number fields, J. London Math. Soc. (2) 16 (1977) 203–208. [Ger78] F. Gerth III, The Hasse norm principle in cyclotomic number fields , J. Reine Angew. Math. 303/304 (1978) 249–252. [Gor80] D. Gorenstein, Finite groups , Second edition, Chelsea Publishing...

    work page arXiv 1977

  7. [7]

    extensions, arXiv:2309.16187. [HY3] A. Hoshi, A. Yamasaki, Hasse norm principle for metacyclic extensions with trivia l Schur multiplier , arXiv:2503.14365. [HY4] A. Hoshi, A. Yamasaki, Hasse norm principle for Heisenberg extensions of degree p3, arXiv:2503.15408. [Hul96] A. Hulpke, Konstruktion transitiver Permutationsgruppen , Ph.D Thesis. Rheinisch-W e...

    work page arXiv

  8. [8]

    Hulpke, Constructing Transitive Permutation Groups , J

    [Hul05] A. Hulpke, Constructing Transitive Permutation Groups , J. Symbolic Comput. 39 (2005) 1–30. [H¨ ur84] W. H¨ urlimann, On algebraic tori of norm type , Comment. Math. Helv. 59 (1984) 539–549. [Jon65] A. Jones, On representations of finite groups over valuation rings , Illinois J. Math. 9 (1965) 297–303. [Kan12] M. Kang, Retract rational fields , J. A...

    work page 2005

  9. [9]

    [KMRT98] M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The book of involutions , American Mathematical Society Colloquium Publications, 44, American Mathematical Socie ty, Providence, RI, 1998, xxii+593 pp. [Kun84] B. E. Kunyavskii, Arithmetic properties of three-dimensional algebraic tor i, (Russian) Integral lattices and fi- nite linear groups, Zap. N...

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  10. [10]

    Macedo, The Hasse norm principle for An-extensions, J

    [Mac20] A. Macedo, The Hasse norm principle for An-extensions, J. Number Theory 211 (2020) 500–512. [Mac25] A. Macedo, On the obstruction to the Hasse principle for multinorm equa tions, Israel J. Math. 267 (2025) 1–33. [MN22] A. Macedo, R. Newton, Explicit methods for the Hasse norm principle and applicati ons to An and Sn extensions, Math. Proc. Cambrid...

    work page 2020

  11. [11]

    North-Holland M athematical Library 4, North-Holland, Amsterdam,

    North-Holland Publi shing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1974, vii+292 pp. North-Holland M athematical Library 4, North-Holland, Amsterdam,

    work page 1974

  12. [12]

    [Man86] Yu. I. Manin, Cubic forms: algebra, geometry, arithmetic , Second edition. North-Holland Mathematical Library 4, North-Holland Publishing Co., Amsterdam, 1986, x+326 pp . [MT86] Yu. I. Manin, M. A. Tsfasman, Rational varieties: algebra, geometry, arithmetic , (Russian) Uspekhi Mat. Nauk 41 (1986) 43–94; translation in Russian Math. Surveys 41 (198...

    work page 1986

  13. [13]

    [Mer20] A. S. Merkurjev, Versal torsors and retracts , Transform. Groups 25 (2020) 843–858. [Mil86] Milne, J. S. Arithmetic duality theorems , Perspect. Math., 1, Academic Press, Inc., Boston, MA, 1986 , x+421 pp. [Miy71] T. Miyata, Invariants of certain groups I , Nagoya Math. J. 41 (1971) 69–73. [NSW00] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology o...

    work page doi:10.57723/289559 2020

  14. [14]

    Pollio, On the multinorm principle for finite abelian extensions , Pure and Applied Mathematics Quarterly 10 (2014) 547–566

    [Pol14] T. Pollio, On the multinorm principle for finite abelian extensions , Pure and Applied Mathematics Quarterly 10 (2014) 547–566. [PR13] T. Pollio, A. S. Rapinchuk, The multinorm principle for linearly disjoint Galois exten sions, J. Number Theory 133 (2013) 802–821. [PR10] G. Prasad, A. S. Rapinchuk, Local-global principles for embedding of fields wi...

    work page 2014

  15. [15]

    [Roy87] G. F. Royle, The transitive groups of degree twelve , J. Symbolic Comput. 4 (1987) 255–268. [Sal84] D. J. Saltman, Retract rational fields and cyclic Galois extensions , Israel J. Math. 47 (1984) 165–215. [Sal99] D. J. Saltman, Lectures on division algebras , CBMS Regional Conference Series in Mathematics, 94, Publi shed by American Mathematical So...

    work page 1987

  16. [16]

    [Swa60] R. G. Swan, Induced Representations and Projective Modules , Ann. Math. 71 (1960) 552–578. [Swa83] R. G. Swan, Noether’s problem in Galois theory , Emmy Noether in Bryn Mawr (Bryn Mawr, Pa., 1982), 21–40, Springer, New York-Berlin,

    work page 1960

  17. [17]

    Tate, Global class field theory , Algebraic Number Theory (Proc

    [Tat67] J. Tate, Global class field theory , Algebraic Number Theory (Proc. Instructional Conf., Brig hton, 1965), 162–203, Thompson, W ashington, D.C.,

    work page 1965

  18. [18]

    [Vos67] V. E. Voskresenskii, On two-dimensional algebraic tori II , (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967) 711–716; translation in Math. USSR-Izv. 1 (1967) 691–696. [Vos69] V. E. Voskresenskii, The birational equivalence of linear algebraic groups , (Russian) Dokl. Akad. Nauk SSSR 188 (1969) 978–981; erratum, ibid. 191 1969 nos., 1, 2, 3, vii;...

    work page 1967

  19. [19]

    [VK84] V. E. Voskresenskii, B. E. Kunyavskii, Maximal tori in semisimple algebraic groups , Kuibyshev State Inst., Kuiby- shev (1984). Deposited in VINITI March 5, 1984, No. 1269-84 D ep. (Ref. Zh. Mat. (1984), 7A405 Dep.). [W at79] W. C. W aterhouse, Introduction to affine group schemes , Grad. Texts in Math., 66 Springer-Verlag, New York- Berlin,

    work page 1984

  20. [20]

    [W ei14] D

    xi+164 pp. [W ei14] D. W ei, On the equation NK/k (Ξ) = P (t). Proc. Lond. Math. Soc. (3) 109 (2014) 1402–1434. [Yam12] A. Yamasaki, Negative solutions to three-dimensional monomial Noether problem, J. Algebra 370 (2012) 46–78. [Zas35] H. Zassenhaus, ¨Uber endliche Fastk¨ orper, (German) Abh. Math. Sem. Univ. Hamburg 11 (1935) 187–220. Universit´e Paris C...

    work page 2014