{"paper":{"title":"Definability and decidability for rings of integers in totally imaginary fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.NT","authors_text":"Caleb Springer","submitted_at":"2022-07-01T01:36:52Z","abstract_excerpt":"We show that the ring of integers of $\\mathbb{Q}^{\\text{tr}}$ is existentially definable in the ring of integers of $\\mathbb{Q}^{\\text{tr}}(i)$, where $\\mathbb{Q}^{\\text{tr}}$ denotes the field of all totally real numbers. This implies that the ring of integers of $\\mathbb{Q}^{\\text{tr}}(i)$ is undecidable and first-order non-definable in $\\mathbb{Q}^{\\text{tr}}(i)$. More generally, when $L$ is a totally imaginary quadratic extension of a totally real field $K$, we use the unit groups $R^\\times$ of orders $R\\subseteq \\mathcal{O}_L$ to produce existentially definable totally real subsets $X\\sub"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.00140","kind":"arxiv","version":3},"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/2207.00140/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"}