Aouissi, Siham; Ismaili, Moulay Chrif; Talbi, Mohamed; Azizi, Abdelmalek Fields \(\mathbb{Q}(\sqrt[3]{d}, \zeta_3)\) whose \(3\)-class group is of type \((9, 3)\). (English) Zbl 1479.11185 Int. J. Number Theory 15, No. 7, 1437-1447 (2019). Summary: Let \(\mathrm{k} = \mathbb{Q}(\sqrt[3]{d}, \zeta_3)\) with \(d\) a cube-free positive integer. Let \(C_{\mathrm{k} , 3}\) be the 3-class group of k. With the aid of genus theory, arithmetic properties of the pure cubic field \(\mathbb{Q}(\sqrt[3]{d})\) and some results on the 3-class group \(C_{\mathrm{k} , 3}\), we determine all integers \(d\) such that \(C_{\mathrm{k} , 3} \simeq \mathbb Z / 9 \mathbb Z \times \mathbb{Z} / 3 \mathbb{Z}\). Cited in 3 Documents 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 11R37 Class field theory Keywords:pure cubic fields; 3-class groups; structure of groups Software:PARI/GP PDFBibTeX XMLCite \textit{S. Aouissi} et al., Int. J. Number Theory 15, No. 7, 1437--1447 (2019; Zbl 1479.11185) Full Text: DOI arXiv References: [1] S. Aouissi, D. C. Mayer, M. C. Ismaili, M. Talbi and A. Azizi, \(3\)-rank of ambiguous class groups in cubic Kummer extensions; arXiv:1804.00767v3. · Zbl 1474.11186 [2] S. Aouissi, M. C. Ismaili, M. Talbi and A. Azizi, The generators of \(3\)-class group of some fields of degree \(6\) over \(\Bbb Q\); arXiv:1804.00692. · Zbl 1474.11185 [3] S. Aouissi, M. Talbi, M. C. Ismaili and A. Azizi, On a conjecture of lemmermeyer; arXiv:1810.07172. · Zbl 1481.11106 [4] Barrucand, P. and Cohn, H., A rational genus, class number divisibility, and unit theory for pure cubic fields, J. Number Theory2 (1970) 7-21. · Zbl 0192.40001 [5] Barrucand, P. and Cohn, H., Remarks on principal factors in a relative cubic field, J. Number Theory3 (1971) 226-239. · Zbl 0218.12002 [6] Beach, B. D., Williams, H. C. and Zarnke, C. R., Some computer results on units in quadratic and cubic fields, in Proc. twenty-fifth Summer Meeting of the Canadian Mathematical Congress (Lake Head University, Thunder Bay, 1971), pp. 609-648. · Zbl 0348.12003 [7] Calegari, F. and Emerton, M., On the ramification of Hecke algebras at Eisenstein primes, Invent. Math.160 (2005) 97-144. · Zbl 1145.11314 [8] Dedekind, R., Über die Anzahl der Idealklassen in reinen kubischen Zahlkörpern, J. Reine Angew. Math.121 (1900) 40-123. · JFM 30.0198.02 [9] Gerth, F. III, On \(3\)-class groups of pure cubic fields, J. Reine Angew. Math.278/279 (1975) 52-62. · Zbl 0334.12011 [10] Gerth, F. III, On \(3\)-class groups of cyclic cubic extensions of certain number fields, J. Number Theory8 (1976) 84-94. · Zbl 0329.12006 [11] Gerth, F. III, On \(3\)-class groups of certain pure cubic fields, Aust. Math. Soc.72(3) (2005) 471-476. · Zbl 1159.11043 [12] Honda, T., Pure cubic fields whose class numbers are multiples of three, J. Number Theory3 (1971) 7-12. · Zbl 0222.12004 [13] Ireland, K. and Rosen, M., A classical introduction to modern number theory, Chapter 9, Cubic and Biquadratic Reciprocity, 2nd edition (Springer, New York, 1992), pp. 108-111. · Zbl 0712.11001 [14] M. C. Ismaili, Sur la capitulation des \(3\)-classes d’idéaux de la clôture normale d’un corps cubique pur, Thèse de doctorat, University of Laval, Québec (1992). [15] Ishida, M., The genus fields of algebraic number fields, Chapter 7, The Genus Fields of Pure Number Fields, , Vol. 555 (Springer-Verlag, 1976), pp. 87-93. · Zbl 0353.12001 [16] Markoff, A., Sur les nombres entiers dépendants d’une racine cubique d’un nombre entier ordinaire, Mem. Acad. Imp. Sci. St. Petersbourg VIII38 (1892) 1-37. [17] The PARI Group, PARI/GP, Version 2.9.4, Bordeaux, 2017; http://pari.math.u-bordeaux.fr. 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.