Dihedral CM fields with class number one (Q1965853)

Let \(E\) be a CM field, i.e., a totally imaginary quadratic extension of a totally real number field, and suppose that \(E\) is normal over \(\mathbb Q\). Let \(E^+\) denote the maximal real subfield, and let \(h_E\), \(h_{E^+}\) be the class numbers. The relative class number \(h^-_E= h_E/h_{E^+}\) is known to be an integer. \textit{A. M. Odlyzko} [Invent. Math. 29, 275--286 (1975; Zbl 0299.12010)] proved that there are only a finite number of such fields \(E\) with \(h_E=1\), and \textit{J. Hoffstein} [Invent. Math. 55, 37--47 (1979; Zbl 0474.12009)] showed that in this case \(\deg E\leq 436\). \textit{K. Yamamura} [Math. Comput. 62, 899--921 (1994; Zbl 0798.11046)] determined all fields \(E\) such that \(h_E=1\) and \(G=\text{Gal}(E/\mathbb Q)\) is abelian. The next step to consider is a dihedral group \(G\), and the author obtains a complete enumeration of the fields \(E\) under these assumptions. The possible degrees are \(8, 12, 16, 20\), and 24. There are exactly 43 fields with \(h^-_E=1\) and of these 32 satisfy \(h_E=1\). For the degrees \(8, 12, 16\), the result has been earlier obtained by \textit{S. Louboutin} and \textit{R. Okazaki} [Proc. Lond. Math. Soc. (3) 76, 523--548 (1998; Zbl 0891.11054)]. The paper is a very thorough investigation containing a large number of results both theoretical and numerical.