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Let $k_2^{(1)}$ be the Hilbert 2-class field of $k$ and $k^{(*)}=Q(\\sqrt{p_1},\\sqrt{p_2},\\sqrt 2, i)$ be its genus field. Let $C_{k,2}$ denote the 2-part of the class group of $k$. The unramified abelian extensions of $k$ are $K_1=k(\\sqrt{p_1})$, $K_2=k(\\sqrt{p_2})$, $K_3=k(\\sqrt{2})$ and $k^{(*)}$. Our goal is to study the capitulation problem of the 2-"},"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":"1503.05132","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.NT","submitted_at":"2015-03-17T17:38:22Z","cross_cats_sorted":[],"title_canon_sha256":"bf8edd0a6ce93707861dd423206a9965adabaa1ecfbde2e82e8f7d6bf0187ff5","abstract_canon_sha256":"08b3f3a6a0ab6d6482003d6a1ce42e67b45f12b1ee1a2dedcdfe14920c70fe61"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:22:20.292796Z","signature_b64":"ZZgWcd9lwTkxX64J79OTQN+Z361XIne0d1/ZVmpVGwe+5UuZlH+fq+YmE9GdUz9Kd7q6yA4uyx1H3ZnY4tK4AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f177bb9fd6d63b48b74d46535388e110ebaa0c0c44c881ccedfd1e927f49f01e","last_reissued_at":"2026-05-18T02:22:20.291983Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:22:20.291983Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sur la capitulation des 2-classes d'id\\'eaux du corps Q(\\sqrt{2p_1p_2}, i)","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Abdelkader Zekhnini, Abdelmalek Azizi, Mohammed Taous","submitted_at":"2015-03-17T17:38:22Z","abstract_excerpt":"Let $p_1$ and $p_2$ be two primes such that $p_1\\equiv p_2\\equiv1 \\pmod4$ and at least two of the three elements $\\{(\\frac{2}{p_1}), (\\frac{2}{p_2}), (\\frac{p_1}{p_2})\\}$ are equal to -1. Put $i=\\sqrt{-1}$, $d=2p_1p_2$ and $k =Q(\\sqrt{d}, i)$. Let $k_2^{(1)}$ be the Hilbert 2-class field of $k$ and $k^{(*)}=Q(\\sqrt{p_1},\\sqrt{p_2},\\sqrt 2, i)$ be its genus field. Let $C_{k,2}$ denote the 2-part of the class group of $k$. The unramified abelian extensions of $k$ are $K_1=k(\\sqrt{p_1})$, $K_2=k(\\sqrt{p_2})$, $K_3=k(\\sqrt{2})$ and $k^{(*)}$. 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