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The class number one problem for some non-abelian normal CM-fields

Author(s): Stéphane Louboutin; Ryotaro Okazaki; Michel Olivier
Journal: Trans. Amer. Math. Soc. 349 (1997), 3657-3678.
MSC (1991): Primary 11R29; Secondary 11R21, 11R42, 11M20, 11Y40
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Abstract: Let ${\bf N}$ be a non-abelian normal CM-field of degree $4p,$ $p$ any odd prime. Note that the Galois group of ${\bf N}$ is either the dicyclic group of order $4p,$ or the dihedral group of order $4p.$ We prove that the (relative) class number of a dicyclic CM-field of degree $4p$ is always greater then one. Then, we determine all the dihedral CM-fields of degree $12$ with class number one: there are exactly nine such CM-fields.

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Additional Information:

Stéphane Louboutin
Affiliation: Université de Caen, UFR Sciences, Département de Mathématiques, Esplanade de la paix, 14032 Caen Cedex, France
Email: loubouti@math.unicaen.fr

Ryotaro Okazaki
Affiliation: Doshisha University, Department of Mathematics, Tanabe, Kyoto, 610-03, Japan
Email: rokazaki@doshisha.ac.jp

Michel Olivier
Affiliation: Laboratoire A2X, UMR 99 36, Université Bordeaux I, 351 Cours de la Libération, 33405 Talence Cedex, France
Email: olivier@math.u-bordeaux.fr

DOI: 10.1090/S0002-9947-97-01768-6
PII: S 0002-9947(97)01768-6
Keywords: CM-field, dihedral field, relative class number
Received by editor(s): July 16, 1995
Received by editor(s) in revised form: March 21, 1996
Copyright of article: Copyright 1997, American Mathematical Society

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