{"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 $K$. For a number field $K$, this is a result of Colliot-Th\\'el\\`ene and Van Geel, proved using results on"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.07357","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}