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We use the Hasse norm theorem, Kummer theory, and class field theory to prove this result. We also prove that for any $n\\in \\mathbb{N}$ and any global field $K$ with $(\\text{char}(K), n)=1$, $K^*\\setminus K^{*n}$ is diophantine over $K$. For a number field $K$, this is a result of Colliot-Th\\'el\\`ene and Van Geel, proved using results on"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1710.07357","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-10-19T21:20:25Z","cross_cats_sorted":["math.LO"],"title_canon_sha256":"2b4da8b93282ab2f48057be31b0603b5c544c23453cf5778188be4aefbaaf0d5","abstract_canon_sha256":"75ec20e5ab879ad5ed647c5c88f43d5d69afad3ef9d2ad18e1f076992c975b7b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:43.785959Z","signature_b64":"UZy8xT0dNQLLXhML0JZig0pV6LoZtIf1R67kgXgNsmfifg8ClLZk3d7N1mnYqR4tp2f4StkRDhU8iaKhkORHBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"01ee3f11ca1ee988482f4067247fdcedba2fdae7ee5d7620517f0bfb33ac4e42","last_reissued_at":"2026-05-18T00:01:43.785296Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:43.785296Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Diophantine definability of nonnorms of cyclic extensions of global fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.NT","authors_text":"Travis Morrison","submitted_at":"2017-10-19T21:20:25Z","abstract_excerpt":"We show that for any square-free natural number $n$ and any global field $K$ with $(\\text{char}(K), n)=1$ containing the $n$th roots of unity, the pairs $(x,y)\\in K^*\\times K^*$ such that $x$ is not a norm of $K(\\sqrt[n]{y})/K$ form a diophantine set over $K$. We use the Hasse norm theorem, Kummer theory, and class field theory to prove this result. We also prove that for any $n\\in \\mathbb{N}$ and any global field $K$ with $(\\text{char}(K), n)=1$, $K^*\\setminus K^{*n}$ is diophantine over $K$. 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