© 1996 by London Mathematical Society
© The London Mathematical Society
Determination of all Quaternion Octic CM-Fields with Class Number 2
Université de Caen, U.F.R. Sciences, Département de Mathématiques Esplanade de la Paix, 14032 Caen Cedex, France. E-mail: loubouti{at}math.unicaen.fr
Received 1 March 1993. Revision received 14 November 1994.
It is known that there are only finitely many normal CM-fields with class number one or with given class number (see [9, Theorem 2; 11, Theorem 2]) and J. Hoffstein showed that the degree of any normal CM-field with class number one is less than 436 (see [2, Corollary 2]). Moreover, K. Yamamura has determined all the abelian CM-fields with class number one: there are 172 non-isomorphic such number fields. In a recent paper the author and R. Okazaki moved on to the determination of non-abelian but normal octic CM-fields with class number one. Noticing that their class numbers are always even, they got rid of quaternion octic CM-fields, then they focussed on dihedral octic CM-fields and proved that there are 17 dihedral octic CM-fields with class number one.
The aim of this paper is to get back to the quaternion case: we shall show that there exists exactly one quaternion octic CM-field with class number 2, namely: Q(
) with = –(2 + 2)(3 + 3). Moreover, we shall show that the Hilbert class field of this number field is a normal and non-abelian CM-field of degree 16 with class number one.
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