Aouissi, Siham; Ismaili, Moulay Chrif; Talbi, Mohamed; Azizi, Abdelmalek The generators of 3-class group of some fields of degree 6 over \(\mathbb{Q}\). (English) Zbl 1474.11185 Bol. Soc. Parana. Mat. (3) 39, No. 3, 37-52 (2021). Summary: Let \(k =\mathbb{Q} (\sqrt[3]{p}, \zeta_3)\), where \(p\) is a prime number such that \(p\equiv 1\pmod{9}\), and let \(C_{k,3}\) be the 3-component of the class group of \(k\). In [Bull. Aust. Math. Soc. 72, No. 3, 471–476 (2005; Zbl 1159.11043)], F. Gerth III proves a conjecture made by F. Calegari and M. Emerton [Invent. Math. 160, No. 1, 97–144 (2005; Zbl 1145.11314)] which gives necessary and sufficient conditions for \(C_{k,3}\) to be of rank two. The purpose of the present work is to determine generators of \(C_{k,3}\), whenever it is isomorphic to \(\mathbb{Z}/9\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}\). Cited in 2 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; generators Citations:Zbl 1159.11043; Zbl 1145.11314 Software:PARI/GP PDFBibTeX XMLCite \textit{S. Aouissi} et al., Bol. Soc. Parana. Mat. (3) 39, No. 3, 37--52 (2021; Zbl 1474.11185) Full Text: arXiv Link References: [1] P. Barrucand and H. Cohn, Remarks on principal factors in a relative cubic field,J. Number Theory3(1971), 226-239. · Zbl 0218.12002 [2] F.Calegari and M. Emerton, On the ramification of Hecke algebras at Eisenstein primes, Invent. Math. 160 (2005), 97-144. · Zbl 1145.11314 [3] R. Dedekind, ¨Uber die Anzahl der Idealklassen in reinen kubischen Zahlk¨orpern.J. f¨ur reine und angewandte Mathematik,Bd.121(1900), 40-123. [4] F. Gerth III, On 3-Class Groups of Cyclic Cubic Extensions of Certain Number Fields,J. Number Theory8(1976), 84-94. · Zbl 0329.12006 [5] F. Gerth III, On 3-class groups of pure cubic fields,J. Reine Angew. Math278/279(1975), 52-62. · Zbl 0334.12011 [6] F. Gerth III, On 3-class groups of certain pure cubic fields,Australian Mathematical Society Volume 72, Number 3, December 2005, pp.471-476. · Zbl 1159.11043 [7] G. Gras, Sur les l-classes d’id´eaux dans les extensions cycliques relatives de degr´e premier l, Ann. Inst. Fourier,Grenoble, tome 23, fasc3(1973). · Zbl 0255.12003 [8] H.Hasse, Bericht ¨Uber neuere Untersuchungen und probleme aus der Theorie der algebraischen Zahlk¨oberrper,I, Jber. Deutsch. Math. Verein.35(1926), 1-55. · JFM 52.0150.19 [9] H. Hasse, Newe Begr¨undung Und Verallgemeinerung der Theorie der Normenrest Symbols, Journal f¨ur reine und ang. Math.162(1930), 134-143. [10] C. S. Herz,Construction of class fields, Seminar on Complex Multiplication, Springer-Verlag, New York, 1966. [11] Kenneth Ireland - Michael Rosen:A classical Introduction to Modern Number Theory, Second Edition, Springer 1992. · Zbl 0712.11001 [12] ThePARIGroup,PARI/GP,Version2.9.4,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.