{"paper":{"title":"On decidable algebraic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.LO","authors_text":"Alexandra Shlapentokh, Moshe Jarden","submitted_at":"2015-02-13T04:46:50Z","abstract_excerpt":"We prove the following propositions. Theorem 1: Let $M$ be a subfield of a fixed algebraic closure $\\tilde \\Q$ of $\\Q$ whose existential elementary theory is decidable (resp. primitively decidable). Then, M is conjugate to a recursive (resp. primitive recursive) subfield $L \\subset \\tilde \\Q$.\n  Theorem 2: For each positive integer $e$ there are infinitely many $e$-tuples $\\boldsymbol \\sigma \\in \\Gal(\\Q)^e$ such that the field $\\tilde \\Q( {\\boldsymbol \\sigma})$ -- the fixed field of $\\boldsymbol \\sigma$, is recursive in $\\tilde\\Q$ and its elementary theory is decidable. Moreover, $\\tilde \\Q(\\b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.03885","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":""},"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"}