On the class number one problem for non-normal quartic CM-fields
Let \(N\) be the normal closure of a non-normal quartic number field. Then, \(N\) is a dihedral octic number field which contains four non-normal quartic subfields. Let \(K\) be any one of these four subfields. We prove that \(K\) is a CM-field if and only if \(N\) is a CM-field. In that case, we prove that the relative class number \(h_ K^ -\) of \(K\) is equal to one if and only if \(h_ N^ -\) of \(N\) is equal to one. Moreover, the four non-normal quartic subfields of \(N\) have the same relative class number. Now, thanks to Dedekind zeta function factorizations, we prove that at least one of the Dedekind zeta functions of these four non-normal subfields does not have any Siegel zero. This enables us to get explicit lower bounds on the relative class number of \(K\). Hence, we get huge explicit upper bounds on the discriminants of non-normal quartic CM- fields with relative class number equal to one. Finally, we explain how one can determine all non-normal quartic CM-fields with class number one that contain a given real quadratic number field. For example, we prove that there are six non-normal quartic CM-fields with class number one that contain the real quadratic number field \(\mathbb{Q}(\sqrt{5})\). Let us point out that using this huge upper bound on the discriminants of the non-normal quartic CM-fields with relative class number one, the author and \textit{R. Okazaki} [Acta Arith. (to appear)] have lately managed to determine all non-normal quartic CM-fields and all non-abelian normal octic CM-fields with class number one.
scientific article; zbMATH DE number 589798
scientific article