The class number one problem for the dihedral CM-fields. (English) Zbl 0958.11071

Lefeuvre, Yann; Louboutin, Stéphane

Halter-Koch, Franz (ed.) et al., Algebraic number theory and diophantine analysis. Proceedings of the international conference, Graz, Austria, August 30-September 5, 1998. Berlin: Walter de Gruyter. 249-275 (2000).

In this survey, the authors explain how to find a complete list of normal CM-fields with class number \(1\) and dihedral Galois group. There are three different problems to solve: one has to prove a lower bound for the relative class number of such fields that is good enough to produce a finite list of fields containing those with class number \(1\), then construct these finitely many fields using class field theory, and finally compute their class number. It turns out that there are exactly \(32\) such fields, and all of them have degree at most \(24\).
For the entire collection see [Zbl 0940.00025].

Reviewer: Franz Lemmermeyer (San Marcos)

MSC:

11R29	Class numbers, class groups, discriminants
11R21	Other number fields
11R37	Class field theory
11Y40	Algebraic number theory computations
11R42	Zeta functions and \(L\)-functions of number fields

Keywords:

dihedral extensions; relative class number