{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:3VXBFLB5N5B7WKP2AHQ46Y4XZD","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":"6a4631683068edd8cdc74ce4a13f797a58ea8075e3e930ee98c97491dea7d4c2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-10-05T09:58:03Z","title_canon_sha256":"33939e0bc13c31600dd4a919176eccfd04f757233bb2d88009af3493ef999173"},"schema_version":"1.0","source":{"id":"1210.1693","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.1693","created_at":"2026-05-18T03:43:53Z"},{"alias_kind":"arxiv_version","alias_value":"1210.1693v1","created_at":"2026-05-18T03:43:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.1693","created_at":"2026-05-18T03:43:53Z"},{"alias_kind":"pith_short_12","alias_value":"3VXBFLB5N5B7","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"3VXBFLB5N5B7WKP2","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"3VXBFLB5","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:a8ab5ee0d2daca2ecab67b374d8636a86ac5eb67923e33d3eeef867aa7a00ac2","target":"graph","created_at":"2026-05-18T03:43:53Z","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 $L/K$ be a finite Galois extension of fields with group $\\Gamma$. When $\\Gamma$ is nilpotent, we show that the problem of enumerating all nilpotent Hopf-Galois structures on $L/K$ can be reduced to the corresponding problem for the Sylow subgroups of $\\Gamma$. We use this to enumerate all nilpotent (resp. abelian) Hopf-Galois structures on a cyclic extension of arbitrary finite degree. When $\\Gamma$ is abelian, we give conditions under which every abelian Hopf-Galois structure on $L/K$ has type $\\Gamma$. We also give a criterion on $n$ such that \\emph{every} Hopf-Galois structure on a cycl","authors_text":"Nigel P. Byott","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-10-05T09:58:03Z","title":"Nilpotent and abelian Hopf-Galois structures on field extensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.1693","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:b49afb188d1d5c6f454fa704252949fed55cf9f57a58f691b97ef50119fa1a65","target":"record","created_at":"2026-05-18T03:43:53Z","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":"6a4631683068edd8cdc74ce4a13f797a58ea8075e3e930ee98c97491dea7d4c2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-10-05T09:58:03Z","title_canon_sha256":"33939e0bc13c31600dd4a919176eccfd04f757233bb2d88009af3493ef999173"},"schema_version":"1.0","source":{"id":"1210.1693","kind":"arxiv","version":1}},"canonical_sha256":"dd6e12ac3d6f43fb29fa01e1cf6397c8d1f33ee95db0b0f06652a6fa15ce0c0c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dd6e12ac3d6f43fb29fa01e1cf6397c8d1f33ee95db0b0f06652a6fa15ce0c0c","first_computed_at":"2026-05-18T03:43:53.581668Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:43:53.581668Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YCLkrjW5WBCXu5Q6MINaGDqsRjG/zbOyO2Av2/ma7kDnlJdZEVSLXInljGzc1zKNiXDhYPfynvtW9GkW/U66Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:43:53.582394Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.1693","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b49afb188d1d5c6f454fa704252949fed55cf9f57a58f691b97ef50119fa1a65","sha256:a8ab5ee0d2daca2ecab67b374d8636a86ac5eb67923e33d3eeef867aa7a00ac2"],"state_sha256":"5f6f438c9f0d63eaa2a2194875fb8ddf226bcdcc69d8b65e1123d2ac16eaef59"}