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We first present certain class fields $\\widetilde{K_{N,p,\\mu}^{1,2}}$ of $K$, in the sense of Hilbert, which are generated by Siegel-Ramachandra invariants of $(K_i)_{(Np^{\\mu+1})}$ for $i=1,2$ over $K_{(Np^\\mu)}$ and show that $K_{(Np^{\\mu+1})}=\\widetilde{K_{N,p,\\mu}^{1,2}}$ for almost all $\\mu$."},"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":"1306.6390","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-06-27T01:22:40Z","cross_cats_sorted":[],"title_canon_sha256":"249bc1084bf3b0281680ca14c3d4e5615420fa5772fa2ad12bd4a0b075fdac39","abstract_canon_sha256":"28c072587b52401d62f11cff2b131259e8e323c0ebbf16d4b65a3eae438922e1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:12.251134Z","signature_b64":"bwSwzclUJVCbeGO+5RcirTbW3/Gy6/mYI0FxjpXQoJTWs0ojhY5TmPXT/eVXKvzAdl3GP3Z34ZUcIWr8K9BCCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"61241a0d4fb02ae6ad3e258a92fc2651e477c97660090583d1358aea065cad79","last_reissued_at":"2026-05-18T01:03:12.250653Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:12.250653Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Construction of class fields over imaginary biquadratic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dong Sung Yoon, Ja Kyung Koo","submitted_at":"2013-06-27T01:22:40Z","abstract_excerpt":"Let $K$ be an imaginary biquadratic field and $K_1$, $K_2$ be its imaginary quadratic subfields. For integers $N>0$, $\\mu\\geq 0$ and an odd prime $p$ with $\\gcd(N,p)=1$, let $K_{(Np^\\mu)}$ and $(K_i)_{(Np^\\mu)}$ for $i=1,2$ be the ray class fields of $K$ and $K_i$, respectively, modulo $Np^\\mu$. We first present certain class fields $\\widetilde{K_{N,p,\\mu}^{1,2}}$ of $K$, in the sense of Hilbert, which are generated by Siegel-Ramachandra invariants of $(K_i)_{(Np^{\\mu+1})}$ for $i=1,2$ over $K_{(Np^\\mu)}$ and show that $K_{(Np^{\\mu+1})}=\\widetilde{K_{N,p,\\mu}^{1,2}}$ for almost all $\\mu$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.6390","kind":"arxiv","version":4},"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":"1306.6390","created_at":"2026-05-18T01:03:12.250716+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.6390v4","created_at":"2026-05-18T01:03:12.250716+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.6390","created_at":"2026-05-18T01:03:12.250716+00:00"},{"alias_kind":"pith_short_12","alias_value":"MESBUDKPWAVO","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_16","alias_value":"MESBUDKPWAVONLJ6","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_8","alias_value":"MESBUDKP","created_at":"2026-05-18T12:27:52.871228+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/MESBUDKPWAVONLJ6EWFJF7BGKH","json":"https://pith.science/pith/MESBUDKPWAVONLJ6EWFJF7BGKH.json","graph_json":"https://pith.science/api/pith-number/MESBUDKPWAVONLJ6EWFJF7BGKH/graph.json","events_json":"https://pith.science/api/pith-number/MESBUDKPWAVONLJ6EWFJF7BGKH/events.json","paper":"https://pith.science/paper/MESBUDKP"},"agent_actions":{"view_html":"https://pith.science/pith/MESBUDKPWAVONLJ6EWFJF7BGKH","download_json":"https://pith.science/pith/MESBUDKPWAVONLJ6EWFJF7BGKH.json","view_paper":"https://pith.science/paper/MESBUDKP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.6390&json=true","fetch_graph":"https://pith.science/api/pith-number/MESBUDKPWAVONLJ6EWFJF7BGKH/graph.json","fetch_events":"https://pith.science/api/pith-number/MESBUDKPWAVONLJ6EWFJF7BGKH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MESBUDKPWAVONLJ6EWFJF7BGKH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MESBUDKPWAVONLJ6EWFJF7BGKH/action/storage_attestation","attest_author":"https://pith.science/pith/MESBUDKPWAVONLJ6EWFJF7BGKH/action/author_attestation","sign_citation":"https://pith.science/pith/MESBUDKPWAVONLJ6EWFJF7BGKH/action/citation_signature","submit_replication":"https://pith.science/pith/MESBUDKPWAVONLJ6EWFJF7BGKH/action/replication_record"}},"created_at":"2026-05-18T01:03:12.250716+00:00","updated_at":"2026-05-18T01:03:12.250716+00:00"}