{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:TSB75O3PVNEVXHRXB7S5YITOWG","short_pith_number":"pith:TSB75O3P","schema_version":"1.0","canonical_sha256":"9c83febb6fab495b9e370fe5dc226eb19438c4fa206d633949487026633f99db","source":{"kind":"arxiv","id":"2011.08842","version":1},"attestation_state":"computed","paper":{"title":"Monogenic fields with odd class number Part II: even degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Artane Siad","submitted_at":"2020-11-17T18:58:24Z","abstract_excerpt":"In 1801, Gauss proved that there were infinitely many quadratic fields with odd class number. We generalise this result by showing that there are infinitely many $S_n$-fields of any given even degree and signature that have odd class number. Also, we prove that there are infinitely many fields of any even degree at least $4$ and with at least one real embedding that have units of every signature. To do so, we bound the average number of $2$-torsion elements in the class group, narrow class group, and oriented class group of monogenised fields of even degree (and compute these averages precisel"},"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":"2011.08842","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-11-17T18:58:24Z","cross_cats_sorted":[],"title_canon_sha256":"6cce7a2173ba5ff89a511a7bfbb95eac73371cc58cc73d71416330a0dc7693de","abstract_canon_sha256":"e3edccb0934bc556e543ed775e4a952af2addd2f8b2f2382e47aa1ae1a1bff52"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:52:31.461140Z","signature_b64":"A/FYRUrTjv9qmJ3DZBxVSBMlg1H5VuoNjcNDkrXFWJHCHqamOzd1mWijgrzF6V3N/VhHTiBrETVkWqW9cBLIAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c83febb6fab495b9e370fe5dc226eb19438c4fa206d633949487026633f99db","last_reissued_at":"2026-07-05T01:52:31.460731Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:52:31.460731Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Monogenic fields with odd class number Part II: even degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Artane Siad","submitted_at":"2020-11-17T18:58:24Z","abstract_excerpt":"In 1801, Gauss proved that there were infinitely many quadratic fields with odd class number. We generalise this result by showing that there are infinitely many $S_n$-fields of any given even degree and signature that have odd class number. Also, we prove that there are infinitely many fields of any even degree at least $4$ and with at least one real embedding that have units of every signature. To do so, we bound the average number of $2$-torsion elements in the class group, narrow class group, and oriented class group of monogenised fields of even degree (and compute these averages precisel"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.08842","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2011.08842/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2011.08842","created_at":"2026-07-05T01:52:31.460787+00:00"},{"alias_kind":"arxiv_version","alias_value":"2011.08842v1","created_at":"2026-07-05T01:52:31.460787+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2011.08842","created_at":"2026-07-05T01:52:31.460787+00:00"},{"alias_kind":"pith_short_12","alias_value":"TSB75O3PVNEV","created_at":"2026-07-05T01:52:31.460787+00:00"},{"alias_kind":"pith_short_16","alias_value":"TSB75O3PVNEVXHRX","created_at":"2026-07-05T01:52:31.460787+00:00"},{"alias_kind":"pith_short_8","alias_value":"TSB75O3P","created_at":"2026-07-05T01:52:31.460787+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2511.00322","citing_title":"Counting the number of $n$-periodic $\\mathbb{Z}_{p}$-and $\\mathbb{F}_{p}[t]$-points of a discrete dynamical system with applications from arithmetic statistics, VI","ref_index":39,"is_internal_anchor":false},{"citing_arxiv_id":"2604.16978","citing_title":"Geometry-of-numbers methods over global fields II: Coregular representations","ref_index":43,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TSB75O3PVNEVXHRXB7S5YITOWG","json":"https://pith.science/pith/TSB75O3PVNEVXHRXB7S5YITOWG.json","graph_json":"https://pith.science/api/pith-number/TSB75O3PVNEVXHRXB7S5YITOWG/graph.json","events_json":"https://pith.science/api/pith-number/TSB75O3PVNEVXHRXB7S5YITOWG/events.json","paper":"https://pith.science/paper/TSB75O3P"},"agent_actions":{"view_html":"https://pith.science/pith/TSB75O3PVNEVXHRXB7S5YITOWG","download_json":"https://pith.science/pith/TSB75O3PVNEVXHRXB7S5YITOWG.json","view_paper":"https://pith.science/paper/TSB75O3P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2011.08842&json=true","fetch_graph":"https://pith.science/api/pith-number/TSB75O3PVNEVXHRXB7S5YITOWG/graph.json","fetch_events":"https://pith.science/api/pith-number/TSB75O3PVNEVXHRXB7S5YITOWG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TSB75O3PVNEVXHRXB7S5YITOWG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TSB75O3PVNEVXHRXB7S5YITOWG/action/storage_attestation","attest_author":"https://pith.science/pith/TSB75O3PVNEVXHRXB7S5YITOWG/action/author_attestation","sign_citation":"https://pith.science/pith/TSB75O3PVNEVXHRXB7S5YITOWG/action/citation_signature","submit_replication":"https://pith.science/pith/TSB75O3PVNEVXHRXB7S5YITOWG/action/replication_record"}},"created_at":"2026-07-05T01:52:31.460787+00:00","updated_at":"2026-07-05T01:52:31.460787+00:00"}