{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:XGDNHCCMY3PV3CHMWBFGOATGMV","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"84579b5a834a89dc45d553de1267fa3219ecc1a7f1638d1e411950e11dabe460","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-11-23T20:19:18Z","title_canon_sha256":"153a33cade4acf09d6560c7a899d0214ad5e8f3471e600420b7d66e1e3a04429"},"schema_version":"1.0","source":{"id":"1111.5604","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.5604","created_at":"2026-05-18T04:07:43Z"},{"alias_kind":"arxiv_version","alias_value":"1111.5604v1","created_at":"2026-05-18T04:07:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.5604","created_at":"2026-05-18T04:07:43Z"},{"alias_kind":"pith_short_12","alias_value":"XGDNHCCMY3PV","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"XGDNHCCMY3PV3CHM","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"XGDNHCCM","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:d1ed26262041458f22c6645b4385484d1a89b4243681da26a08b56bac3e9e99c","target":"graph","created_at":"2026-05-18T04:07:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let D be a division ring. We say that D is left algebraic over a (not necessarily central) subfield K of D if every x in D satisfies a polynomial equation x^n + a_{n-1}x^{n-1}+...+a_0=0 with a_0,...,a_{n-1} in K. We show that if D is a division ring that is left algebraic over a subfield K of bounded degree d then D is at most d^2-dimensional over its center. This generalizes a result of Kaplansky. For the proof we give a new version of the combinatorial theorem of Shirshov that sufficiently long words over a finite alphabet contain either a q-decomposable subword or a high power of a non-triv","authors_text":"Jason P. Bell, Vesselin Drensky, Yaghoub Sharifi","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-11-23T20:19:18Z","title":"Shirshov's theorem and division rings that are left algebraic over a subfield"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.5604","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:610f99dd65180f20e282bd34c94f70c4b9e5242e0267ec90cf2e76f3f6f416b8","target":"record","created_at":"2026-05-18T04:07:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"84579b5a834a89dc45d553de1267fa3219ecc1a7f1638d1e411950e11dabe460","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-11-23T20:19:18Z","title_canon_sha256":"153a33cade4acf09d6560c7a899d0214ad5e8f3471e600420b7d66e1e3a04429"},"schema_version":"1.0","source":{"id":"1111.5604","kind":"arxiv","version":1}},"canonical_sha256":"b986d3884cc6df5d88ecb04a670266657e670a3038769c9dc2fa04485c1f26c5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b986d3884cc6df5d88ecb04a670266657e670a3038769c9dc2fa04485c1f26c5","first_computed_at":"2026-05-18T04:07:43.397569Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:07:43.397569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qVP9/JGquZh2nyXDoDwlAIv1bBdDwlfBmE8n++JSECGUAbKiAltDwin89zbuv6MOse9wKttXAdVPJChjJHcrBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:07:43.398252Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.5604","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:610f99dd65180f20e282bd34c94f70c4b9e5242e0267ec90cf2e76f3f6f416b8","sha256:d1ed26262041458f22c6645b4385484d1a89b4243681da26a08b56bac3e9e99c"],"state_sha256":"ada0f24dcbf619dfd35b4c91eb135ca909e571604b526d8f9c2bb58c1bc6a451"}