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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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The class number one problem for some non-abelian normal CM-fields
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by Stéphane Louboutin, Ryotaro Okazaki and Michel Olivier PDF
Trans. Amer. Math. Soc. 349 (1997), 3657-3678 Request permission

Abstract:

Let $\textbf {N}$ be a non-abelian normal CM-field of degree $4p,$ $p$ any odd prime. Note that the Galois group of $\textbf {N}$ 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
  • Received by editor(s): July 16, 1995
  • Received by editor(s) in revised form: March 21, 1996
  • © Copyright 1997 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 349 (1997), 3657-3678
  • MSC (1991): Primary 11R29; Secondary 11R21, 11R42, 11M20, 11Y40
  • DOI: https://doi.org/10.1090/S0002-9947-97-01768-6
  • MathSciNet review: 1390044