×

On a conjecture of Lemmermeyer. (English) Zbl 1481.11106

Authors’ abstract: Let \(p \equiv 1\pmod{3}\) be a prime and denote by \(\zeta_3\) a primitive third root of unity. Recently, Lemmermeyer presented a conjecture about \(3\)-class groups of pure cubic fields \(L = \mathbb Q (\sqrt[3]{p})\) and of their normal closures \(\text{k} = \mathbb Q (\sqrt[3]{p}, \zeta_3)\). The main goal of this paper is to reduce Lemmermeyer’s conjecture to a problem of unit theory by showing that the conjecture of Lemmermeyer follows from Conjecture 2.9.
Reviewer’s remarks: Lemmermeyer’s conjecture is (of course) stated in the paper under review, but somewhat lengthy to be reproduced here. Instead, I write down the text of Conjecture 2.9 that the authors do present. Namely,
Conjecture 2.9: Let \(k=\mathbb{Q}(\sqrt[3]{p}, \zeta_3)\), where \(p\) is a prime number such that \(p\equiv 4\text{ or }7\pmod 9\). Then \[ \Biggl(\frac{3}{p}\Biggr)_3= 1\Rightarrow u= 3, \] where \((\frac{}{p})_3\) is the cubic residue symbol, and \(u=[E_k:E_0]\) is the index of the subgroup generated by units of the intermediate fields of the extension \(k/\mathbb{Q}\) in \(E_k\), the group of units in \(k\).
As the authors remark, Conjecture 2.9 did occur in a few explicitly mentioned sources by them, for example by F. Gerth III, as Case 3, page 175 in [Bull. Aust. Math. Soc. 72, No. 3, 471–476 (2005; Zbl 1159.11043)].
The paper itself regarding the techniques involved, is given in full detail. Worthwhile to be studied.

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

Citations:

Zbl 1159.11043

Software:

PARI/GP
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aouissi, S., Ismaili, M. C., Talbi, M. and Azizi, A., The generators of \(3\)-class group of some fields of degree \(6\) over \(\Bbb Q\), Boletim Sociedade Paranaense de Mathematica Journal, https://doi.org/10.5269/bspm.40672. · Zbl 1479.11185
[2] Aouissi, S., Mayer, D. C., Ismaili, M. C., Talbi, M. and Azizi, A., \(3\)-rank of ambiguous class groups in cubic Kummer extensions, Periodica Mathematica Hungarica Journal, https://doi.org/10.1007/s10998-020-00326-1.
[3] Aouissi, S., Talbi, M., Ismaili, M. C. and Azizi, A., Fields \(\Bbb Q(\sqrt[ 3]{ d}, \zeta_3)\) whose \(3\)-class group is of type \((9,3)\), Int. J. Number Theory15(7) (2019) 1437-1447. · Zbl 1479.11185
[4] Barrucand, P. and Cohn, H., A rational genus, class number divisibility, and unit theory for pure cubic fields, J. Number Theory2(1) (1970) 7-21. · Zbl 0192.40001
[5] Barrucand, P. and Cohn, H., Remarks on principal factors in a relative cubic field, J. Number Theory3(2) (1971) 226-239. · Zbl 0218.12002
[6] Barrucand, P., Williams, H. C. and Baniuk, L., A computational technique for determining the class number of a pure cubic field, Math. Comp.30(134) (1976) 312-323. · Zbl 0324.12005
[7] 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 Congr. (Lake Head University, Thunder Bay, 1971), pp. 609-648. · Zbl 0348.12003
[8] Calegari, F. and Emerton, M., On the ramification of Hecke algebras at Eisenstein primes, Invent. Math.160 (2005) 97-144. · Zbl 1145.11314
[9] Dedekind, R., Über die Anzahl der Idealklassen in reinen kubischen Zahlkörpern, J. Reine Angew. Math.121 (1900) 40-123. · JFM 30.0198.02
[10] R. El Masaoudi, Sur la divisibilité exacte par \(3\) du nombre de classes d’un corps cubique pur et le problème de capitulation, Thèse de doctorat, Université Mohammed premier, Oujda (2002).
[11] Gerth, F. III, On \(3\)-class groups of pure cubic fields, J. Reine Angew. Math278/279 (1975) 52-62 · Zbl 0334.12011
[12] Gerth, F. III, On \(3\)-class groups of cyclic cubic extensions of certain number fields, J. Number Theory8(1) (1976) 84-98. · Zbl 0329.12006
[13] Gerth, F. III, On \(3\)-class groups of certain pure cubic fields, Bull. Austral. Math. Soc.72 (2005) 471-476. · Zbl 1159.11043
[14] Gras, G., Sur les \(\ell \)-classes d’idéaux des extensions non galoisiennes de \(\mathbf{Q}\) de degré premier impair \(\ell\) a cloture galoisienne diédrale de degré \(2\ell \), J. Math. Soc. Japan26 (1974) 677-685. · Zbl 0279.12004
[15] H. Hasse, The class number formula of Dedekind-Meyer for simply real cubic fields, unpublished manuscript.
[16] Herz, C. S., Construction of class fields, Chap. 7, in Seminar on Complex MultiplicationInstitute Advanced Study, Princeton, 1957-1958, ed. Borel, A.et al. Vol. 21, (Springer-Verlag, New York, 1966).
[17] Honda, T., Pure cubic fields whose class numbers are multiples of three, J. Number Theory3(1) (1971) 7-12. · Zbl 0222.12004
[18] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics Vol. 84, (Springer-Verlag, New York, 1982). · Zbl 0482.10001
[19] M. C. Ismaili, Sur la capitulation des \(3\)-classes d’idéaux de la clôture normale d’un corps cubique pur, Thèse de doctorat, Universite Laval, Québec (1992).
[20] Ismaili, M. C. and El Mesaoudi, R., Corps cubiques purs dont le nombre de classes est exactement divisible par \(3\), Ann. Sci Math. Québec28(1-2) (2004) 103-112. · Zbl 1104.11046
[21] F. Lemmermeyer, Class field towers, in Unsolved Problems, Chap. 1 (2010), p. 44.
[22] Lemmermeyer, F., Reciprocity Laws: From Euler to Eisenstein (Springer Science and Business Media, 2013). · Zbl 0949.11002
[23] Markoff, A. A., Sur les nombres entiers dépendants d’une racine cubique d’un nombre entier ordinaire, Mém. Acad. Imp. Sci. St. Pétersbourg (Sér. VII)38(9) (1892) 1-37.
[24] Meyer, C., Die Berechnung der Klassenzahl Abelscher Zahlkörper über quadratischen Zahlkörpern (Akademie Verlag, Berlin, 1957). · Zbl 0079.06001
[25] Mollin, R. A., Algebraic Number Theory, (Chapman and Hall, London, UK, 1999). · Zbl 0930.11001
[26] PARI Developer Group, PARI/GP, Version 2.9.4, Bordeaux (2017), http://pari.math.u-bordeaux.fr.
[27] Williams, H. C., Determination of principal factors in \(\Bbb Q(\sqrt{ D})\) and \(\Bbb Q(\sqrt[ 3]{ D})\), Math. Comp.38(157) (1982) 261-274.
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.