Bicubic number fields with large class numbers. (English) Zbl 1460.11135

Morita, Yasuko; Umegaki, Atsuki; Umegaki, Yumiko

Mishou, Hidehiko (ed.) et al., Various aspects of multiple zeta functions – in honor of Professor Kohji Matsumoto’s 60th birthday. Proceedings of the international conference, Nagoya University, Nagoya, Japan August 21–25, 2020. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 84, 335-351 (2020).

Summary: For a given finite group \(G\), the problem whether there exist infinitely many number fields \(K\) with large class number and Galois group \(\text{Gal}(K/\boldsymbol{\text{Q}}) \cong G\) is interesting and important. This problem was proved affirmatively for some groups \(G\).
In this paper, we approach this problem by considering \(h_KR_K\), where \(h_K\) is the class number of \(K\) and \(R_K\) is the regulator of \(K\). We prove that there exist infinitely many bicubic number fields \(K\) with large \(h_KR_K\). Moreover, we also prove generalization of the claim.
For the entire collection see [Zbl 1446.11004].

MSC:

11R29	Class numbers, class groups, discriminants
11R21	Other number fields

Keywords:

class number; bicubic number fields

References:

[1]	Joseph H. Silverman, An inequality relating the regulator and the discriminant of a number fields,J. Number Theory19(1984), no. 3 437-442. · Zbl 0552.12003
[2]	Atsuki Umegaki and Yumiko Umegaki, Quartic fields with large class number, to appear in Rocky Mountain J. · Zbl 1448.11201