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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"},"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":"1502.03885","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-02-13T04:46:50Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"e403f5b42bc12a5fc34b38919ff6ed12837433c91c112d9a5490e0c2fc68abaf","abstract_canon_sha256":"95590961e0c56e70b690d1f079da2df1aae00dbed788e8ccc6ff424010172c35"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:07.031886Z","signature_b64":"o1oP3S5zUzdrD0UTIC9qLiHJkBm9KJmZ9QrHo5hP4rmrfMd8h1c507VvYnkBdw2pLD9DTqhUvXugwMf1UYQWCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"20c52315ff2944b7102d09e9a701b7dbe7800751eb7dc00d2316447bdd03219b","last_reissued_at":"2026-05-18T02:27:07.031093Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:07.031093Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1502.03885","created_at":"2026-05-18T02:27:07.031218+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.03885v1","created_at":"2026-05-18T02:27:07.031218+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.03885","created_at":"2026-05-18T02:27:07.031218+00:00"},{"alias_kind":"pith_short_12","alias_value":"EDCSGFP7FFCL","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_16","alias_value":"EDCSGFP7FFCLOEBN","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_8","alias_value":"EDCSGFP7","created_at":"2026-05-18T12:29:19.899920+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/EDCSGFP7FFCLOEBNBHU2OANX3P","json":"https://pith.science/pith/EDCSGFP7FFCLOEBNBHU2OANX3P.json","graph_json":"https://pith.science/api/pith-number/EDCSGFP7FFCLOEBNBHU2OANX3P/graph.json","events_json":"https://pith.science/api/pith-number/EDCSGFP7FFCLOEBNBHU2OANX3P/events.json","paper":"https://pith.science/paper/EDCSGFP7"},"agent_actions":{"view_html":"https://pith.science/pith/EDCSGFP7FFCLOEBNBHU2OANX3P","download_json":"https://pith.science/pith/EDCSGFP7FFCLOEBNBHU2OANX3P.json","view_paper":"https://pith.science/paper/EDCSGFP7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.03885&json=true","fetch_graph":"https://pith.science/api/pith-number/EDCSGFP7FFCLOEBNBHU2OANX3P/graph.json","fetch_events":"https://pith.science/api/pith-number/EDCSGFP7FFCLOEBNBHU2OANX3P/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EDCSGFP7FFCLOEBNBHU2OANX3P/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EDCSGFP7FFCLOEBNBHU2OANX3P/action/storage_attestation","attest_author":"https://pith.science/pith/EDCSGFP7FFCLOEBNBHU2OANX3P/action/author_attestation","sign_citation":"https://pith.science/pith/EDCSGFP7FFCLOEBNBHU2OANX3P/action/citation_signature","submit_replication":"https://pith.science/pith/EDCSGFP7FFCLOEBNBHU2OANX3P/action/replication_record"}},"created_at":"2026-05-18T02:27:07.031218+00:00","updated_at":"2026-05-18T02:27:07.031218+00:00"}