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

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].


11R29 Class numbers, class groups, discriminants
11R21 Other number fields
Full Text: DOI Euclid


[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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.