{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:DZQKVEHROZ7DKCGCUYNESQDA3A","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":"8b58f45af6e7e88980a0149cf149a07970f3d572c6a77a165d22ccf4a57bdbc4","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-08-17T01:50:48Z","title_canon_sha256":"bd7c4f8bd9237b1afa28cfd62745b7822eb3edccf91a3f33a8aa08c59da6b1d0"},"schema_version":"1.0","source":{"id":"1108.3379","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.3379","created_at":"2026-05-18T04:05:47Z"},{"alias_kind":"arxiv_version","alias_value":"1108.3379v1","created_at":"2026-05-18T04:05:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.3379","created_at":"2026-05-18T04:05:47Z"},{"alias_kind":"pith_short_12","alias_value":"DZQKVEHROZ7D","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"DZQKVEHROZ7DKCGC","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"DZQKVEHR","created_at":"2026-05-18T12:26:26Z"}],"graph_snapshots":[{"event_id":"sha256:04d5f6d418a91466caacbd46594252a7405bc7eb12176676c323062e400face4","target":"graph","created_at":"2026-05-18T04:05:47Z","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 $G$ be a finite group and $k$ be a field. Let $G$ act on the rational function field $k(x_g:g\\in G)$ by $k$-automorphisms defined by $g\\cdot x_h=x_{gh}$ for any $g,h\\in G$. Noether's problem asks whether the fixed field $k(G)=k(x_g:g\\in G)^G$ is rational (i.e. purely transcendental) over $k$. Theorem 1. If $G$ is a group of order $2^n$ ($n\\ge 4$) and of exponent $2^e$ such that (i) $e\\ge n-2$ and (ii) $\\zeta_{2^{e-1}} \\in k$, then $k(G)$ is $k$-rational. Theorem 2. Let $G$ be a group of order $4n$ where $n$ is any positive integer (it is unnecessary to assume that $n$ is a power of 2). Ass","authors_text":"Ivo M. Michailov, Jian Zhou, Ming-chang Kang","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-08-17T01:50:48Z","title":"Noether's problem for the groups with a cyclic subgroup of index 4"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.3379","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":"sha256:32928cd693393c682ce0812888001a44146a225fe9657ee1a54bea438c686b83","target":"record","created_at":"2026-05-18T04:05:47Z","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":"8b58f45af6e7e88980a0149cf149a07970f3d572c6a77a165d22ccf4a57bdbc4","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-08-17T01:50:48Z","title_canon_sha256":"bd7c4f8bd9237b1afa28cfd62745b7822eb3edccf91a3f33a8aa08c59da6b1d0"},"schema_version":"1.0","source":{"id":"1108.3379","kind":"arxiv","version":1}},"canonical_sha256":"1e60aa90f1767e3508c2a61a494060d827b937d113eabefc237cb8d14fea5b7d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1e60aa90f1767e3508c2a61a494060d827b937d113eabefc237cb8d14fea5b7d","first_computed_at":"2026-05-18T04:05:47.176508Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:05:47.176508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LP6IivZ/MjYlkV8xyWMDNA7wT3EubeGQGXUCFF0kM2qzH3QPF12ssCzN3uV7Mi+eCIhnu8nWNEYwK9by3ZKvDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:05:47.176899Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.3379","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:32928cd693393c682ce0812888001a44146a225fe9657ee1a54bea438c686b83","sha256:04d5f6d418a91466caacbd46594252a7405bc7eb12176676c323062e400face4"],"state_sha256":"705a6b3211ef95d4d739a17485ca0b85ab96bc60c66409e46f1e6a4d9f2ad81a"}