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

##### 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