{"paper":{"title":"A Dedekind's Criterion over Valued Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Abdulaziz Deajim, Lhoussain El fadil, Mhammed Boulagouaz","submitted_at":"2019-08-18T02:50:20Z","abstract_excerpt":"Let $(K,\\nu)$ be an arbitrary-rank valued field, $R_\\nu$ its valuation ring, $K(\\alpha)/K$ a separable finite field extension generated over $K$ by a root of a monic irreducible polynomial $f\\in R_\\nu[X]$. We give necessary and sufficient conditions for $R_\\nu[\\alpha]$ to be integrally closed. We further characterize the integral closedness of $R_\\nu[\\alpha]$ based on information about the valuations on $K(\\alpha)$ extending $\\nu$. Our results enhance and generalize some existing results in the relevant literature. Some applications and examples are also given."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.06365","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.06365/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}