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Let $\\mathbf{k}=\\mathbb{Q}(\\sqrt{2p}, i)$, $\\mathbf{k}_1^{(2)}$ 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 $\\mathbf{k}_2^{(2)}/\\mathbf{k}$. Suppose that the $2$-part, $C_{\\mathbf{k}, 2}$, of the class group of $\\mathbf{k}$ is of type $(2, 4)$; then $\\mathbf{k}_1^{(2)}$ contains six extensions $\\mathbf{K_{i, j}}/\\mathbf{k}$, $i=1, 2, 3$ and $j=2, 4$. Our goal is to st"},"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":"1402.1228","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-02-06T02:07:58Z","cross_cats_sorted":[],"title_canon_sha256":"72c0cd5290ebe2a04ebba05c57260920a915d05cea3f7ce21ac125dfdb7ddbc9","abstract_canon_sha256":"ae9ed7ef7e0f365d7053d1c73b1f3c2be686e81b2b0ad5125a3c3f243f419cb2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:44:47.157617Z","signature_b64":"j/TCVuHuY1rL7MJ3HS38kd3lBaZLfTPzKotLtZwtmhWDpOQzVWxypcq0zsHHHgGU+MHzqfG04oMvWxdEk+L9BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ef47738e50ccab503e0a43549cb00066f6c80557c86d83cbc4b9e7a1c10de2e4","last_reissued_at":"2026-05-18T01:44:47.156945Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:44:47.156945Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Capitulation des 2-classes d'id\\'eaux de $\\mathbf{k}=\\mathbb{Q}(\\sqrt{2p}, i)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Abdelmalek Azizi, Mohammed Taous","submitted_at":"2014-02-06T02:07:58Z","abstract_excerpt":"Let $p$ be a prime number such that $p\\equiv 1$ mod $8$ and $i=\\sqrt{-1}$. Let $\\mathbf{k}=\\mathbb{Q}(\\sqrt{2p}, i)$, $\\mathbf{k}_1^{(2)}$ 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 $\\mathbf{k}_2^{(2)}/\\mathbf{k}$. Suppose that the $2$-part, $C_{\\mathbf{k}, 2}$, of the class group of $\\mathbf{k}$ is of type $(2, 4)$; then $\\mathbf{k}_1^{(2)}$ contains six extensions $\\mathbf{K_{i, j}}/\\mathbf{k}$, $i=1, 2, 3$ and $j=2, 4$. 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