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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$","authors_text":"Aiichi Yamasaki, Akinari Hoshi, Mathieu Florence","cross_cats":["math.NT"],"headline":"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.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-05-17T12:48:44Z","title":"Rationality problem for norm one tori of tensor products of \\'etale algebras and Hasse norm principle"},"references":{"count":20,"internal_anchors":1,"resolved_work":20,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"[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-R","work_id":"fed8f766-6b49-495e-a165-3fbc369ec000","year":1984},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"Beauville, The L¨ uroth problem, Rationality problems in algebraic geometry, 1–27, Lectur e Notes in Math., 2172, Fond","work_id":"a3688e6b-c2e7-47fd-babb-902f672feea9","year":2016},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"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 T","work_id":"c7f9845c-065b-401f-bf9a-2f5cd9853f70","year":1990},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Colliot-Th´ el` ene, A","work_id":"83fa3c35-1f61-41b7-8542-c865156b8988","year":2021},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"[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 fo","work_id":"55e52796-a03e-4fe5-b03c-1afe0027679f","year":2014}],"snapshot_sha256":"1f31d87432dadc3a9a1513c172141a662053176a5115aa2b20fee427437ef5a2"},"source":{"id":"2605.17427","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T22:46:23.779974Z","id":"11a612fc-c6c9-4735-b02d-e1b706deb954","model_set":{"reader":"grok-4.3"},"one_line_summary":"Under a coprimeness condition on extension degrees, stable or retract rationality of norm one tori is preserved under tensor product, implying the Hasse norm principle holds for the combined extension over global fields.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"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.","strongest_claim":"If gcd(m_i, n_j | 1≤i≤r, 1≤j≤s)=1 and T_A and T_B are stably (resp. retract) k-rational, then the algebraic k-torus T_A ⊗ T_B and the norm one torus T_{A⊗B} are stably (resp. retract) k-rational. In particular, if k is a global field, then the Hasse norm principle holds for (A⊗B)/k.","weakest_assumption":"The coprimeness condition gcd of all degrees m_i and n_j equals 1 is required for the rationality preservation to hold under tensor product; the abstract presents this as a necessary hypothesis for the stated theorem."}},"verdict_id":"11a612fc-c6c9-4735-b02d-e1b706deb954"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:4de7fc991cbe0963b21e5812c10740afe8de56649296c22da33e3591592516fa","target":"record","created_at":"2026-05-20T00:03:57Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"430dd87f0928ccded7a5ea1528ef8dd8c6a0c6675109ba8ac4ee18584eb05656","cross_cats_sorted":["math.NT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-05-17T12:48:44Z","title_canon_sha256":"d3dc4cdfca003f9d01d6b53b351417db14367a1cde5c78ccfe51c105c00b0d39"},"schema_version":"1.0","source":{"id":"2605.17427","kind":"arxiv","version":1}},"canonical_sha256":"5a959ecd21bb0671fd1c74bc95909b7eb1313805cc9574fc6be9fe39f08ed2fc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5a959ecd21bb0671fd1c74bc95909b7eb1313805cc9574fc6be9fe39f08ed2fc","first_computed_at":"2026-05-20T00:03:57.988067Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:03:57.988067Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7RF4ojqnX7jZCNzNbCSrk7tuZR1YRdH1peDjtCvQCmYsTzNdHulHKezk54YhdLvPhBpiukFSpSgUEIPeKpARCQ==","signature_status":"signed_v1","signed_at":"2026-05-20T00:03:57.988882Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.17427","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4de7fc991cbe0963b21e5812c10740afe8de56649296c22da33e3591592516fa","sha256:b567c0002efb65c5a65ba9593568ae107958588175c701b9f43b87735d3d3e72"],"state_sha256":"3124597e2239beeede3f763a23e5335f7739f3c7d52c2ce38b89e8adb003f00d"}