Dihedral and cyclic extensions with large class numbers
Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 3, p. 583-603

This paper is a continuation of [2]. We construct unconditionally several families of number fields with large class numbers. They are number fields whose Galois closures have as the Galois groups, dihedral groups D n , n=3,4,5, and cyclic groups C n , n=4,5,6. We first construct families of number fields with small regulators, and by using the strong Artin conjecture and applying some modification of zero density result of Kowalski-Michel, we choose subfamilies such that the corresponding L-functions are zero free close to 1. For these subfamilies, the L-functions have the extremal value at s=1, and by the class number formula, we obtain large class numbers.

Cet article est la suite de [2]. Nous construisons inconditionnellement plusieurs familles de corps de nombres ayant un grand nombre de classes. Ce sont des corps de nombres dont la clôture galoisienne a pour groupe de Galois les groupes dièdraux D n , n=3,4,5, et les groupes cycliques C n , n=4,5,6. Nous construisons d’abord des familles de corps de nombres à petits régulateurs et, en utilisant la conjecture d’Artin forte et en appliquant une variante du résultat de densité nulle de Kowalski et Michel, nous choisissons des sous-familles telles que les fonctions L correspondantes soient sans zéro près de 1. Pour ces sous-familles, la fonction L prend une valeur extrémale en s=1 et, par la formule du nombre de classes, nous obtenons un grand nombre de classes.

@article{JTNB_2012__24_3_583_0,
     author = {Cho, Peter J. and Kim, Henry H.},
     title = {Dihedral and cyclic extensions with large class numbers},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {3},
     year = {2012},
     pages = {583-603},
     doi = {10.5802/jtnb.812},
     mrnumber = {3010630},
     zbl = {1275.11145},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2012__24_3_583_0}
}
Cho, Peter J.; Kim, Henry H. Dihedral and cyclic extensions with large class numbers. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 3, pp. 583-603. doi : 10.5802/jtnb.812. http://www.numdam.org/item/JTNB_2012__24_3_583_0/

[1] A. Baker, Linear forms in the logarithms of algebraic numbers I. Mathematika 13 (1966), 204–216. | MR 258756 | Zbl 0161.05201

[2] P. J. Cho and H. H. Kim, Application of the strong Artin conjecture to class number problem. To appear in Can. J. Math.

[3] D. Cox, Galois Theory. Wiley, 2004. | MR 2119052 | Zbl 1057.12002

[4] W. Duke, Number fields with large class groups. Number theory, 117–126, CRM Proc. Lecture Notes, 36, Amer. Math. Soc., Providence, RI, 2004. | MR 2076589 | Zbl 1104.11050

[5] R. C. Daileda, Non-abelian number fields with very large class numbers. Acta Arith. 125 (2006) no.3 , 215–255. | MR 2276192 | Zbl 1158.11044

[6] R. C. Daileda, R. Krishnamoorthy, and A. Malyshev, Maximal class numbers of CM number fields. J. Number Theory 130 (2010), 936–943. | MR 2600412 | Zbl 1185.11065

[7] M. Drmota and M. Skalba, On multiplicative and linear independence of polynomial roots. Contributions to general algebra 7, Vienna, 1991, 127–135. | MR 1143074 | Zbl 0757.12001

[8] V. Ennola, S. Maki, and T. Turunen, On real cyclic sextic fields Math. Comp. Vol 45 (1985), 591–611. | MR 804948 | Zbl 0581.12003

[9] M.-N. Gras, Special Units in Real Cyclic sextic Fields. Math. Comp. 48 (1987) no.177, 179–182. | MR 866107 | Zbl 0617.12006

[10] C. Hooley, Applications of sieve methods to the theory of numbers. Cambridge; New York: Cambridge University Press, 1976. | MR 404173 | Zbl 0327.10044

[11] M. Ishida, Fundamental Units of Certain Number Fields. Abh. Math. Sem. Univ. Hamburg. 39 (1973), 245–250. | MR 335469 | Zbl 0311.12005

[12] S. Jeannin, Nombre de classes et unités des corps de nombres cycliques quintiques d’E.Lehmer. J. Theor. Nombres Bordeaux 8 (1996), no. 1, 75–92. | Numdam | MR 1399947 | Zbl 0865.11070

[13] C. Jensen, A. Ledet and N. Yui, Generic Polynomials; Constructive Aspects of the Inverse Galois Problem. Mathematical Sciences Research Institute Publications 45, Cambridge University Press 2002. | MR 1969648 | Zbl 1042.12001

[14] E. Kowalski and P. Michel, Zeros of families of automorphic L-functions close to 1. Pacific J. Math. 207 (2002), no.2, 411–431. | MR 1972253 | Zbl 1129.11316

[15] M. J. Lavallee, B. K. Spearman, K. S. Williams and Q. Yang, Dihedral Quintic Fields With A Power Basis. Math J. Okayama Univ. 47 (2005), 75–79. | MR 2198862 | Zbl 1161.11393

[16] A. J. Lazarus, The Class Numbers and Cyclotomy of Simplest Quartic Fields. PhD thesis, University of California, Berkeley, 1989. | MR 2638618

[17] E. Lehmer, Connection between Gaussian periods and cyclic units. Math. Comp. 50 (1988) no. 182, 341–355. | MR 929551 | Zbl 0652.12004

[18] M. Nair, Power Free Values of Polynomials. Mathematika 23 (1976), 159–183. | MR 429801 | Zbl 0349.10039

[19] K. Nakamula, Certain Quartic Fields with Small Regulators. J. Number Theory 57 (1996), no.1 , 1–21. | MR 1378570 | Zbl 0847.11057

[20] J. Sandor, D. S. Mitrinovic and B. Crstici, Handbook of Number Theory I. 1st edn, Springer-Netherlands (1995). | MR 2186914 | Zbl 1151.11300

[21] R. Schoof and L. Washington, Polynomials and Real Cyclotomic Fields with Large Class Number. Math. Comp. 50 (1988) no. 182, 543–556. | MR 929552 | Zbl 0649.12007

[22] A.M. Schöpp, Fundamental units in a parametric family of not totally real quintic number fields. J. de Theorie des Nom. de Bordeaux, 18 (2006), no. 3, 693–706. | Numdam | MR 2330436 | Zbl 1119.11065

[23] J.P. Serre, Topics in Galois Theory. Research Notes in Mathematics, A K Peters, Ltd. 2008. | MR 2363329 | Zbl 1128.12001

[24] Y.-Y. Shen, Unit groups and class numbers of real cyclic octic fields. Trans. Amer. Math. Soc. 326 (1991), no. 1, 179–209. | MR 1031243 | Zbl 0738.11055

[25] J.H. Silverman, An inequality relating the regulator and the discriminant of a number field. J. Number Theory 19 (1984), no. 3, 437–442. | MR 769793 | Zbl 0552.12003

[26] L.C. Washington, Introduction to Cyclotomic Fields. Graduate Texts in Math., vol 83, Springer-Verlag, New York, 1982. | MR 718674 | Zbl 0484.12001

[27] L.C. Washington, Class numbers of the simplest cubic fields. Math. Comp. 48 (1987), no. 177, 371–384. | MR 866122 | Zbl 0613.12002