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Dihedral CM fields with class number one. (Corps diédraux à multiplication complexe principaux.) (French) Zbl 0952.11024

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. 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 J. Hoffstein [Invent. Math. 55, 37–47 (1979; Zbl 0474.12009)] showed that in this case \(\deg E\leq 436\).
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 S. Louboutin and 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.
Reviewer: V.Ennola (Turku)

MSC:

11R29 Class numbers, class groups, discriminants
11R37 Class field theory
11R21 Other number fields
11R42 Zeta functions and \(L\)-functions of number fields
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