{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:YL5UXN2ZHOXBLSSXWAFJKLBZIY","short_pith_number":"pith:YL5UXN2Z","canonical_record":{"source":{"id":"1811.05535","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-11-13T21:38:00Z","cross_cats_sorted":[],"title_canon_sha256":"20005d91f046ce14fde4fffad5e225ca4620e431b110aaadf544f52dfe9fec5a","abstract_canon_sha256":"e152a7afcacd5523ae07033fca09ce5773bfa8a63091b4193c849d8ba9aa229a"},"schema_version":"1.0"},"canonical_sha256":"c2fb4bb7593bae15ca57b00a952c39463f2a74e22979051010cf2e4de9da7b4c","source":{"kind":"arxiv","id":"1811.05535","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.05535","created_at":"2026-05-18T00:00:42Z"},{"alias_kind":"arxiv_version","alias_value":"1811.05535v1","created_at":"2026-05-18T00:00:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.05535","created_at":"2026-05-18T00:00:42Z"},{"alias_kind":"pith_short_12","alias_value":"YL5UXN2ZHOXB","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"YL5UXN2ZHOXBLSSX","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"YL5UXN2Z","created_at":"2026-05-18T12:33:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:YL5UXN2ZHOXBLSSXWAFJKLBZIY","target":"record","payload":{"canonical_record":{"source":{"id":"1811.05535","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-11-13T21:38:00Z","cross_cats_sorted":[],"title_canon_sha256":"20005d91f046ce14fde4fffad5e225ca4620e431b110aaadf544f52dfe9fec5a","abstract_canon_sha256":"e152a7afcacd5523ae07033fca09ce5773bfa8a63091b4193c849d8ba9aa229a"},"schema_version":"1.0"},"canonical_sha256":"c2fb4bb7593bae15ca57b00a952c39463f2a74e22979051010cf2e4de9da7b4c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:42.703469Z","signature_b64":"umR6c+Xe/bUSbpgk9LkaBcxECazHfkog6io1mgayEaj3bfywTub8zsT/CKWyVVr+5VkpkkTwnfGjakRsQz3oCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c2fb4bb7593bae15ca57b00a952c39463f2a74e22979051010cf2e4de9da7b4c","last_reissued_at":"2026-05-18T00:00:42.703037Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:42.703037Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1811.05535","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:00:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"apcthXjtA9D5PjEYTCuhTlzYQ0nJVHg0kB2uxh/iIjuq2hLXiUpt+A8alK/T8/37Awhj6KdzQkFq62FgR7bHDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T14:57:37.029987Z"},"content_sha256":"cdc8a6b6c7e38e1614e711e3977bbaa09d75ba9699ee3f47cd872cea12aa3858","schema_version":"1.0","event_id":"sha256:cdc8a6b6c7e38e1614e711e3977bbaa09d75ba9699ee3f47cd872cea12aa3858"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:YL5UXN2ZHOXBLSSXWAFJKLBZIY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Classes of order 4 in the strict class group of number fields and remarks on unramified quadratic extensions of unit type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"David S. Dummit","submitted_at":"2018-11-13T21:38:00Z","abstract_excerpt":"Let $K$ be a number field of degree $n$ over ${\\mathbb Q}$. Then the 4-rank of the strict class group of $K$ is at least ${\\text{rank}_2 \\, } ({ E_{K}^{+} } / E_K^2) - \\lfloor n /2 \\rfloor$ where $E_K$ and ${ E_{K}^{+} }$ denote the units and the totally positive units of $K$, respectively, and $\\text{rank}_2$ is the dimension as an elementary abelian 2-group. In particular, the strict class group of a totally real field $K$ with a totally positive system of fundamental units contains at least$(n-1)/2$ ($n$ odd) or $n/2 -1$ ($n$ even) independent elements of order 4. We also investigate when u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.05535","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:00:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2/pubq6qb/nemr56Mi3aUWtKMhFL16ylGewnnpTMycyK23VDhgrk+Nv9JMbCjQHln+Ok5NkXPL0Fv5Sf5tSYDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T14:57:37.030344Z"},"content_sha256":"8a9ce36515053d6d25f54805573431670297f11151c0adefa702b72a591440f9","schema_version":"1.0","event_id":"sha256:8a9ce36515053d6d25f54805573431670297f11151c0adefa702b72a591440f9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YL5UXN2ZHOXBLSSXWAFJKLBZIY/bundle.json","state_url":"https://pith.science/pith/YL5UXN2ZHOXBLSSXWAFJKLBZIY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YL5UXN2ZHOXBLSSXWAFJKLBZIY/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-27T14:57:37Z","links":{"resolver":"https://pith.science/pith/YL5UXN2ZHOXBLSSXWAFJKLBZIY","bundle":"https://pith.science/pith/YL5UXN2ZHOXBLSSXWAFJKLBZIY/bundle.json","state":"https://pith.science/pith/YL5UXN2ZHOXBLSSXWAFJKLBZIY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YL5UXN2ZHOXBLSSXWAFJKLBZIY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:YL5UXN2ZHOXBLSSXWAFJKLBZIY","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":"e152a7afcacd5523ae07033fca09ce5773bfa8a63091b4193c849d8ba9aa229a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-11-13T21:38:00Z","title_canon_sha256":"20005d91f046ce14fde4fffad5e225ca4620e431b110aaadf544f52dfe9fec5a"},"schema_version":"1.0","source":{"id":"1811.05535","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.05535","created_at":"2026-05-18T00:00:42Z"},{"alias_kind":"arxiv_version","alias_value":"1811.05535v1","created_at":"2026-05-18T00:00:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.05535","created_at":"2026-05-18T00:00:42Z"},{"alias_kind":"pith_short_12","alias_value":"YL5UXN2ZHOXB","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"YL5UXN2ZHOXBLSSX","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"YL5UXN2Z","created_at":"2026-05-18T12:33:04Z"}],"graph_snapshots":[{"event_id":"sha256:8a9ce36515053d6d25f54805573431670297f11151c0adefa702b72a591440f9","target":"graph","created_at":"2026-05-18T00:00:42Z","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 $K$ be a number field of degree $n$ over ${\\mathbb Q}$. Then the 4-rank of the strict class group of $K$ is at least ${\\text{rank}_2 \\, } ({ E_{K}^{+} } / E_K^2) - \\lfloor n /2 \\rfloor$ where $E_K$ and ${ E_{K}^{+} }$ denote the units and the totally positive units of $K$, respectively, and $\\text{rank}_2$ is the dimension as an elementary abelian 2-group. In particular, the strict class group of a totally real field $K$ with a totally positive system of fundamental units contains at least$(n-1)/2$ ($n$ odd) or $n/2 -1$ ($n$ even) independent elements of order 4. We also investigate when u","authors_text":"David S. Dummit","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-11-13T21:38:00Z","title":"Classes of order 4 in the strict class group of number fields and remarks on unramified quadratic extensions of unit type"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.05535","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:cdc8a6b6c7e38e1614e711e3977bbaa09d75ba9699ee3f47cd872cea12aa3858","target":"record","created_at":"2026-05-18T00:00:42Z","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":"e152a7afcacd5523ae07033fca09ce5773bfa8a63091b4193c849d8ba9aa229a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-11-13T21:38:00Z","title_canon_sha256":"20005d91f046ce14fde4fffad5e225ca4620e431b110aaadf544f52dfe9fec5a"},"schema_version":"1.0","source":{"id":"1811.05535","kind":"arxiv","version":1}},"canonical_sha256":"c2fb4bb7593bae15ca57b00a952c39463f2a74e22979051010cf2e4de9da7b4c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c2fb4bb7593bae15ca57b00a952c39463f2a74e22979051010cf2e4de9da7b4c","first_computed_at":"2026-05-18T00:00:42.703037Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:00:42.703037Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"umR6c+Xe/bUSbpgk9LkaBcxECazHfkog6io1mgayEaj3bfywTub8zsT/CKWyVVr+5VkpkkTwnfGjakRsQz3oCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:00:42.703469Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.05535","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cdc8a6b6c7e38e1614e711e3977bbaa09d75ba9699ee3f47cd872cea12aa3858","sha256:8a9ce36515053d6d25f54805573431670297f11151c0adefa702b72a591440f9"],"state_sha256":"ecb07e9d3279c16b679dc16b39c2f6ce87987422ba5f0d9444369e14385b4cbd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K7dReSxthXdxwxoZOqbwtZWiS+T30ACJAJ2D9b1fggTwzQRSnhD8HH1rCo1+5Z2zFMWnPbmBT+M2C1KObA65Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T14:57:37.032429Z","bundle_sha256":"08092042c7c0bb5122b21f394252cefb0292cb5af43d520c8084eb8f89ecba73"}}