{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:SKVJTNSZZFE3ZG4ELAEZ7YKGR3","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":"5ace5723c60e5f91b8bc71b2e647518d50b356b72d32cf9ad3123350f35786a9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.NT","submitted_at":"2015-03-12T06:24:13Z","title_canon_sha256":"8528210401b3745dcfd82b618863d76d3830e369fcf3232f77a98585ddd83320"},"schema_version":"1.0","source":{"id":"1503.03604","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.03604","created_at":"2026-05-18T02:24:27Z"},{"alias_kind":"arxiv_version","alias_value":"1503.03604v1","created_at":"2026-05-18T02:24:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.03604","created_at":"2026-05-18T02:24:27Z"},{"alias_kind":"pith_short_12","alias_value":"SKVJTNSZZFE3","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"SKVJTNSZZFE3ZG4E","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"SKVJTNSZ","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:538eb7e351b75afb18927828a209674c28b97029441a326626df951626a5068f","target":"graph","created_at":"2026-05-18T02:24:27Z","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 $p_1 \\equiv p_2 \\equiv5\\pmod8$ be different primes. Put $i=\\sqrt{-1}$ and $d=2p_1p_2$, then the bicyclic biquadratic field $k=Q(\\sqrt{d}, \\sqrt{-1})$ has an elementary abelian 2-class group of rank $3$. In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-abelian Galois group $\\mathrm{Gal}(k_2^{(2)}/k)$ of the second Hilbert 2-class field $k_2^{(2)}$ of $k$. We study the capitulation problem of the 2-classes of $k$ in its seven unramified quadratic extensions $K_i$ and in its seven unramified bicyclic biquadratic extensions $L_i$.","authors_text":"Abdelkader Zekhnini, Abdelmalek Azizi, Mohammed Taous","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.NT","submitted_at":"2015-03-12T06:24:13Z","title":"Structure of $Gal(k_2^{(2)}/k)$ for some fields $k=Q(\\sqrt{ 2p_1p_2}, \\sqrt{-1})$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.03604","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":"sha256:76c2f4c44e17c2302c7bf35a4c0f9f6e274511d13cec067d80285ae376de0443","target":"record","created_at":"2026-05-18T02:24:27Z","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":"5ace5723c60e5f91b8bc71b2e647518d50b356b72d32cf9ad3123350f35786a9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.NT","submitted_at":"2015-03-12T06:24:13Z","title_canon_sha256":"8528210401b3745dcfd82b618863d76d3830e369fcf3232f77a98585ddd83320"},"schema_version":"1.0","source":{"id":"1503.03604","kind":"arxiv","version":1}},"canonical_sha256":"92aa99b659c949bc9b8458099fe1468ecb0cdfba9dd4177ec36922ed8fa9fc9b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"92aa99b659c949bc9b8458099fe1468ecb0cdfba9dd4177ec36922ed8fa9fc9b","first_computed_at":"2026-05-18T02:24:27.921698Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:24:27.921698Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tE30xnoHV10HAYLa9IndlF0bGKPLtxu8C5Q1gBsAU3KmY238B24HovK9Z16nMsMdCFJAsmwUyjHRpd5WQ8yQAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:24:27.922422Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.03604","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:76c2f4c44e17c2302c7bf35a4c0f9f6e274511d13cec067d80285ae376de0443","sha256:538eb7e351b75afb18927828a209674c28b97029441a326626df951626a5068f"],"state_sha256":"a4bb37e5b94071f5cc2cf0da9710513820d25ff53167b7facab48e3382987ff3"}