{"paper":{"title":"Structure of $Gal(k_2^{(2)}/k)$ for some fields $k=Q(\\sqrt{ 2p_1p_2}, \\sqrt{-1})$","license":"http://creativecommons.org/licenses/publicdomain/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Abdelkader Zekhnini, Abdelmalek Azizi, Mohammed Taous","submitted_at":"2015-03-12T06:24:13Z","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 $L_i$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.03604","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"}