Dihedral and cyclic extensions with large class numbers
The authors construct unconditionally families of number fields with large class numbers whose Galois closures have as Galois groups: dihedral groups \(D_n\), where \(n=3,4,5\), or cyclic groups \(C_ n\), where \(n=4,5,6\). They first construct families of number fields with small regulators, and by using the strong Artin conjecture and applying some modification of 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. This paper is a continuation of \textit{P. J. Cho} and \textit{H. H. Kim} [Can. J. Math. 65, No. 6, 1201--1216 (2013; Zbl 1283.11155)].
scientific article; zbMATH DE number 6142425
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