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Let $\\mathbf{k}_1^{(2)}$ be the Hilbert $2$-class field of $\\mathbf{k}$, $\\mathbf{k}_2^{(2)}$ be the Hilbert $2$-class field of $\\mathbf{k}_1^{(2)}$ and $G=\\mathrm{Gal}(\\mathbf{k}_2^{(2)}/\\mathbf{k})$ be the Galois group of $\\mathbf{k}_2^{(2)}/\\mathbf{k}$. Our goal is to give necessary and sufficient conditions to have $G$ metacyclic in the case where $d=pq$, with $p$ and $q$ are primes such that $p\\equiv 1\\pmod 8$ and $q\\equiv 5\\pmod 8$ or $p\\equiv 1\\pmod 8$ and $q\\equiv 3\\pmod 4$."},"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":"1403.0662","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-03-04T02:19:31Z","cross_cats_sorted":[],"title_canon_sha256":"ed0e027b2da9bb7b96dfcc727a2f7d3894552b45d7df82a3eab0fdd3bbe9e0f4","abstract_canon_sha256":"8e2eafa897949d3388153023ff9246f3e6d124ed03b2522306c5e2e0a3e62df8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:16.015822Z","signature_b64":"9fFHs1UijknG0lN9OipsuSRC2MRHWbf9Vr6kas0a12KNao/AeoVfZnzZnCMx+1HO8PoIoQpsl2jL/xnnJFDHAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b3ac1daf45c0ca6c55fd217a5ff5c127e142b318b098aba9347bd298b82e4371","last_reissued_at":"2026-05-18T02:57:16.015167Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:16.015167Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the rank of the $2$-class group of $\\mathbb{Q}(\\sqrt{p}, \\sqrt{q},\\sqrt{-1})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Abdelkader Zekhnini, Abdelmalek Azizi Mohammed Taous","submitted_at":"2014-03-04T02:19:31Z","abstract_excerpt":"Let $d$ be a square-free integer, $\\mathbf{k}=\\mathbb{Q}(\\sqrt d,\\,i)$ and $i=\\sqrt{-1}$. Let $\\mathbf{k}_1^{(2)}$ be the Hilbert $2$-class field of $\\mathbf{k}$, $\\mathbf{k}_2^{(2)}$ be the Hilbert $2$-class field of $\\mathbf{k}_1^{(2)}$ and $G=\\mathrm{Gal}(\\mathbf{k}_2^{(2)}/\\mathbf{k})$ be the Galois group of $\\mathbf{k}_2^{(2)}/\\mathbf{k}$. Our goal is to give necessary and sufficient conditions to have $G$ metacyclic in the case where $d=pq$, with $p$ and $q$ are primes such that $p\\equiv 1\\pmod 8$ and $q\\equiv 5\\pmod 8$ or $p\\equiv 1\\pmod 8$ and $q\\equiv 3\\pmod 4$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.0662","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1403.0662","created_at":"2026-05-18T02:57:16.015268+00:00"},{"alias_kind":"arxiv_version","alias_value":"1403.0662v1","created_at":"2026-05-18T02:57:16.015268+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.0662","created_at":"2026-05-18T02:57:16.015268+00:00"},{"alias_kind":"pith_short_12","alias_value":"WOWB3L2FYDFG","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_16","alias_value":"WOWB3L2FYDFGYVP5","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_8","alias_value":"WOWB3L2F","created_at":"2026-05-18T12:28:54.890064+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WOWB3L2FYDFGYVP5EF5F75OBE7","json":"https://pith.science/pith/WOWB3L2FYDFGYVP5EF5F75OBE7.json","graph_json":"https://pith.science/api/pith-number/WOWB3L2FYDFGYVP5EF5F75OBE7/graph.json","events_json":"https://pith.science/api/pith-number/WOWB3L2FYDFGYVP5EF5F75OBE7/events.json","paper":"https://pith.science/paper/WOWB3L2F"},"agent_actions":{"view_html":"https://pith.science/pith/WOWB3L2FYDFGYVP5EF5F75OBE7","download_json":"https://pith.science/pith/WOWB3L2FYDFGYVP5EF5F75OBE7.json","view_paper":"https://pith.science/paper/WOWB3L2F","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1403.0662&json=true","fetch_graph":"https://pith.science/api/pith-number/WOWB3L2FYDFGYVP5EF5F75OBE7/graph.json","fetch_events":"https://pith.science/api/pith-number/WOWB3L2FYDFGYVP5EF5F75OBE7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WOWB3L2FYDFGYVP5EF5F75OBE7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WOWB3L2FYDFGYVP5EF5F75OBE7/action/storage_attestation","attest_author":"https://pith.science/pith/WOWB3L2FYDFGYVP5EF5F75OBE7/action/author_attestation","sign_citation":"https://pith.science/pith/WOWB3L2FYDFGYVP5EF5F75OBE7/action/citation_signature","submit_replication":"https://pith.science/pith/WOWB3L2FYDFGYVP5EF5F75OBE7/action/replication_record"}},"created_at":"2026-05-18T02:57:16.015268+00:00","updated_at":"2026-05-18T02:57:16.015268+00:00"}