Dihedral and cyclic extensions with large class numbers

Abstract

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