Application of the strong Artin conjecture to the class number problem (Q2862940)

The authors construct unconditionally families of number fields \(K\) with the largest possible class number, when the degree of \(K/\mathbb{Q}\) is \(4\) or \(5\), and the Galois closure of \(K\) has as Galois group: \(A_4\), \(S_4\), or \(S_5\) . They first construct families of number fields with small regulators, and by using the strong Artin conjecture and applying a zero density result in [\textit{E. Kowalski} and \textit{P. Michel}, Pac. J. Math. 207, No. 2, 411--431 (2002; Zbl 1129.11316)] they 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, they obtain large class numbers. The method of the proof of the results is analogous to that in [\textit{P. J. Cho} and \textit{H. H. Kim}, J. Théor. Nombres Bordx. 24, No. 3, 583--603 (2012; Zbl 1275.11145)].