## 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].

### MSC:

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

### Keywords:

class number; bicubic number fields
Full Text:

### 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
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