{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:RA2DENABBYCIRYETNINKAB3GR5","short_pith_number":"pith:RA2DENAB","canonical_record":{"source":{"id":"1404.3761","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-14T21:37:07Z","cross_cats_sorted":[],"title_canon_sha256":"5f31e2da17e41b79f25c097c7be307b9c55b9b1b2c0adbf1942b8776bc8dafed","abstract_canon_sha256":"f4d2318246fa7cd209ce2c0500be6ecf7330c5d8fcbbf34f7a16d4ed67d14549"},"schema_version":"1.0"},"canonical_sha256":"88343234010e0488e0936a1aa007668f6095dbf4145f8541b26c565d79364060","source":{"kind":"arxiv","id":"1404.3761","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1404.3761","created_at":"2026-05-18T02:54:13Z"},{"alias_kind":"arxiv_version","alias_value":"1404.3761v1","created_at":"2026-05-18T02:54:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.3761","created_at":"2026-05-18T02:54:13Z"},{"alias_kind":"pith_short_12","alias_value":"RA2DENABBYCI","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"RA2DENABBYCIRYET","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"RA2DENAB","created_at":"2026-05-18T12:28:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:RA2DENABBYCIRYETNINKAB3GR5","target":"record","payload":{"canonical_record":{"source":{"id":"1404.3761","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-14T21:37:07Z","cross_cats_sorted":[],"title_canon_sha256":"5f31e2da17e41b79f25c097c7be307b9c55b9b1b2c0adbf1942b8776bc8dafed","abstract_canon_sha256":"f4d2318246fa7cd209ce2c0500be6ecf7330c5d8fcbbf34f7a16d4ed67d14549"},"schema_version":"1.0"},"canonical_sha256":"88343234010e0488e0936a1aa007668f6095dbf4145f8541b26c565d79364060","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:13.705437Z","signature_b64":"AIC6uCfQU7Zu7lPfQxbZbn3oKQJ58vC+ydFhH56IcRpzATRVLTG87wIhztPMdR9mw3e1nzNexlhphLinJmLIDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"88343234010e0488e0936a1aa007668f6095dbf4145f8541b26c565d79364060","last_reissued_at":"2026-05-18T02:54:13.704897Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:13.704897Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1404.3761","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":"2026-05-18T02:54:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Zptg3sHIXEawjgq/Tftx1rNgiJDWjwcQQZPhwauBpOZaze2Q2dDPlKGO+AB6OaD/mgSUDdZEsQ6KsKCkMr+3BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T21:46:10.925535Z"},"content_sha256":"0fed9a9739af6ace8d85d0a870a0cf7db7e297049882c830f9bddeb33e9232d4","schema_version":"1.0","event_id":"sha256:0fed9a9739af6ace8d85d0a870a0cf7db7e297049882c830f9bddeb33e9232d4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:RA2DENABBYCIRYETNINKAB3GR5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Principalization of $2$-class groups of type $(2,2,2)$ of biquadratic fields $\\mathbb{Q}\\left(\\sqrt{\\strut p_1p_2q},\\sqrt{\\strut -1}\\right)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Abdelkader Zekhnini, Abdelmalek Azizi, Daniel C. Mayer, Mohammed Taous","submitted_at":"2014-04-14T21:37:07Z","abstract_excerpt":"Let $p_1\\equiv p_2\\equiv -q\\equiv1 \\pmod4$ be different primes such that $\\displaystyle\\left(\\frac{2}{p_1}\\right)= \\displaystyle\\left(\\frac{2}{p_2}\\right)=\\displaystyle\\left(\\frac{p_1}{q}\\right)=\\displaystyle\\left(\\frac{p_2}{q}\\right)=-1$. Put $d=p_1p_2q$ and $i=\\sqrt{-1}$, then the bicyclic biquadratic field ${k}=\\mathbb{Q}(\\sqrt{d},i)$ has an elementary abelian $2$-class group, $\\mathbf{C}l_2(k)$, of rank $3$. In this paper, we study the principalization of the $2$-classes of ${k}$ in its fourteen unramified abelian extensions $\\mathbb{K}_j$ and $\\mathbb{L}_j$ within ${k}_2^{(1)}$, that is t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.3761","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":"2026-05-18T02:54:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"COBhzdLgsR0K6BxBMUTHyQwQaCoNuuUht7zaOKrIuySJLjg8NmFY+RF5kzsKK6ulZEiiSV4uDBv7VFrN9whiAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T21:46:10.925874Z"},"content_sha256":"963ef1fd236b00267bb1b964e042fedfa03cb0336839e037bda861de60027e96","schema_version":"1.0","event_id":"sha256:963ef1fd236b00267bb1b964e042fedfa03cb0336839e037bda861de60027e96"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RA2DENABBYCIRYETNINKAB3GR5/bundle.json","state_url":"https://pith.science/pith/RA2DENABBYCIRYETNINKAB3GR5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RA2DENABBYCIRYETNINKAB3GR5/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":"2026-06-21T21:46:10Z","links":{"resolver":"https://pith.science/pith/RA2DENABBYCIRYETNINKAB3GR5","bundle":"https://pith.science/pith/RA2DENABBYCIRYETNINKAB3GR5/bundle.json","state":"https://pith.science/pith/RA2DENABBYCIRYETNINKAB3GR5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RA2DENABBYCIRYETNINKAB3GR5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:RA2DENABBYCIRYETNINKAB3GR5","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":"f4d2318246fa7cd209ce2c0500be6ecf7330c5d8fcbbf34f7a16d4ed67d14549","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-14T21:37:07Z","title_canon_sha256":"5f31e2da17e41b79f25c097c7be307b9c55b9b1b2c0adbf1942b8776bc8dafed"},"schema_version":"1.0","source":{"id":"1404.3761","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1404.3761","created_at":"2026-05-18T02:54:13Z"},{"alias_kind":"arxiv_version","alias_value":"1404.3761v1","created_at":"2026-05-18T02:54:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.3761","created_at":"2026-05-18T02:54:13Z"},{"alias_kind":"pith_short_12","alias_value":"RA2DENABBYCI","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"RA2DENABBYCIRYET","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"RA2DENAB","created_at":"2026-05-18T12:28:46Z"}],"graph_snapshots":[{"event_id":"sha256:963ef1fd236b00267bb1b964e042fedfa03cb0336839e037bda861de60027e96","target":"graph","created_at":"2026-05-18T02:54:13Z","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 $p_1\\equiv p_2\\equiv -q\\equiv1 \\pmod4$ be different primes such that $\\displaystyle\\left(\\frac{2}{p_1}\\right)= \\displaystyle\\left(\\frac{2}{p_2}\\right)=\\displaystyle\\left(\\frac{p_1}{q}\\right)=\\displaystyle\\left(\\frac{p_2}{q}\\right)=-1$. Put $d=p_1p_2q$ and $i=\\sqrt{-1}$, then the bicyclic biquadratic field ${k}=\\mathbb{Q}(\\sqrt{d},i)$ has an elementary abelian $2$-class group, $\\mathbf{C}l_2(k)$, of rank $3$. In this paper, we study the principalization of the $2$-classes of ${k}$ in its fourteen unramified abelian extensions $\\mathbb{K}_j$ and $\\mathbb{L}_j$ within ${k}_2^{(1)}$, that is t","authors_text":"Abdelkader Zekhnini, Abdelmalek Azizi, Daniel C. Mayer, Mohammed Taous","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-14T21:37:07Z","title":"Principalization of $2$-class groups of type $(2,2,2)$ of biquadratic fields $\\mathbb{Q}\\left(\\sqrt{\\strut p_1p_2q},\\sqrt{\\strut -1}\\right)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.3761","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":"sha256:0fed9a9739af6ace8d85d0a870a0cf7db7e297049882c830f9bddeb33e9232d4","target":"record","created_at":"2026-05-18T02:54:13Z","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":"f4d2318246fa7cd209ce2c0500be6ecf7330c5d8fcbbf34f7a16d4ed67d14549","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-14T21:37:07Z","title_canon_sha256":"5f31e2da17e41b79f25c097c7be307b9c55b9b1b2c0adbf1942b8776bc8dafed"},"schema_version":"1.0","source":{"id":"1404.3761","kind":"arxiv","version":1}},"canonical_sha256":"88343234010e0488e0936a1aa007668f6095dbf4145f8541b26c565d79364060","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"88343234010e0488e0936a1aa007668f6095dbf4145f8541b26c565d79364060","first_computed_at":"2026-05-18T02:54:13.704897Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:54:13.704897Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AIC6uCfQU7Zu7lPfQxbZbn3oKQJ58vC+ydFhH56IcRpzATRVLTG87wIhztPMdR9mw3e1nzNexlhphLinJmLIDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:54:13.705437Z","signed_message":"canonical_sha256_bytes"},"source_id":"1404.3761","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0fed9a9739af6ace8d85d0a870a0cf7db7e297049882c830f9bddeb33e9232d4","sha256:963ef1fd236b00267bb1b964e042fedfa03cb0336839e037bda861de60027e96"],"state_sha256":"31a64a5c8bc4f57f9fd7d4aec88f4dfdeb549a5552857127d8a83b1562535181"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w6hNuTgx/JgzRtKF5jEkPK8bsXdSRC5VXOG+caXfdDHaOIoOcPSoyl9lZ9hB33iLX2b+nXXq5OUdZslm9yT6DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T21:46:10.927770Z","bundle_sha256":"11854041daa4cecc6cb8e67fbf270ba9fcac63d6bb5b29a003ce66c626e7871a"}}