Azizi, Abdelmalek; Zekhnini, Abdelkader; Taous, Mohammed On the capitulation of the 2-ideal classes of the field \(\mathbb{Q}(\sqrt{p_1p_2q},i)\) of type \((2,2,2)\). (English) Zbl 1431.11118 Bol. Soc. Parana. Mat. (3) 38, No. 4, 127-135 (2020). Summary: We study the capitulation of the 2-ideal classes of the field \(\Bbbk= \mathbb{Q}(\sqrt{p_1p_2q},\sqrt{-1})\), where \(p_1 \equiv p_2 \equiv -q \equiv 1 \pmod{4}\) are different primes, in its three quadratic extensions contained in its absolute genus field \(\Bbbk^\ast\) whenever the 2-class group of \(\Bbbk\) is of type \((2,2,2)\). Cited in 1 Document MSC: 11R11 Quadratic extensions 11R16 Cubic and quartic extensions 11R20 Other abelian and metabelian extensions 11R27 Units and factorization 11R29 Class numbers, class groups, discriminants Keywords:absolute genus fields; fundamental systems of units; 2-class group; capitulation; quadratic fields; biquadratic fields; multiquadratic CM-fields PDFBibTeX XMLCite \textit{A. Azizi} et al., Bol. Soc. Parana. Mat. (3) 38, No. 4, 127--135 (2020; Zbl 1431.11118) Full Text: Link References: [1] A. Azizi,Sur la capitulation des2-classes d’id´eaux dek=Q(√2pq, i), o‘up≡ −q≡1 (mod 4),Acta. Arith. 94, 383-399 (2000). [2] A. Azizi,Unit´es de certains corps de nombres imaginaires et ab´eliens surQ,Ann. Sci. Math. Qu´ebec 23, no 1, 15-21 (1999). [3] A. Azizi and M. Taous,D´etermination des corpsk=Q(d,−1)dont les2-groupes de classes sont de type(2,4)ou(2,2,2), Rend. Istit. Mat. Univ. Trieste.40(2008), 93-116, Zbl 1215.11107, MR2583453. · Zbl 1215.11107 [4] A. Azizi, A. Zekhnini and M. Taous,On the generators of the2-class group of the field k=Q(√d, i), IJPAM, Volume81, No. 5 (2012), 773-784. [5] A. Azizi, A. Zekhnini and M. Taous,On the strongly ambiguous classes ofk/Q(i)where k=Q(√2p1p2, i), Asian-Eur. J. Math.7(2014), no. 1, Zbl 1292.11119, MR3189588. · Zbl 1292.11119 [6] A. Azizi, A. Zekhnini and M. Taous,On the strongly ambiguous classes of some biquadratic number fields,Math. Boh.141, no 3, (2016), 363-384. · Zbl 1413.11120 [7] A. Azizi, A. Zekhnini and M. Taous,Structure ofGal(k(2)2/k)for some fieldsk= Q(√2p1p2, i)withCl2(k)≃(2,2,2), Abh. Math. Sem. Univ. Hamburg, Vol 84,2(2014), 203-231, MR3267742. · Zbl 1361.11063 [8] A. Azizi, A. Zekhnini, M. Taous and Daniel C. Mayer,Principalization of2-class groups of type(2,2,2)of biquadratic fieldsQ(√p1p2q, i), Int. J. Number Theory, DOI: 10.1142/S1793042115500645. · Zbl 1319.11079 [9] A. Azizi, A. Zekhnini and M. Taous,Coclass ofGal(k(2)2/k)for some fieldsk=Q√p1p2q, i with2-class groups of type(2,2,2),J. Algebra Appl, DOI: 10.1142/S0219498816500274. [10] A. Azizi, A. Zekhnini and M. Taous,Capitulation in Abelian extensions of some fields Q(√p1p2q, i),AIP Conf. Proc. 1705 (1) (2016) 1-8. [11] F. Terada,A principal ideal theorem in the genus fields,Tohoku Math. J.23, No. 2 (1971), 697-718, Zbl0243.12003, MR0306158. · Zbl 0243.12003 [12] H. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.