{"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"}