Application of the Strong Artin Conjecture to the Class Number Problem

Abstract

We construct unconditionally several families of number fields with the largest possible class numbers. They are number fields of degree 4 and 5 whose Galois closures have the Galois group A(4) S-4, and S-5. We first construct families of number fields with smallest regulators, and by using the strong Artin conjecture and applying the zero density result of Kowalski-Michel, we choose subfamilies of L-functions that 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 the extreme class numbers