Bicubic number fields with large class numbers

Abstract

For a given finite group $G$, the problem whether there exist infinitely many number fields $K$ with large class number and Galois group $\mathrm{Gal}(K/\boldsymbol{\mathrm{Q}}) \cong G$ 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.