DETERMINATION OF ALL QUATERNION OCTIC CM-FIELDS WITH CLASS NUMBER-2

It is known that there are only finitely many normal CM-fields with cl ass number one or with given class number (see [9, Theorem 2; 11, Theo rem 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]). Moreov er, K. Yamamura has determined all the abelian CM-fields with class nu mber 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 tha t their class numbers are always even, they got rid of quaternion octi c 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 gel back to the quaternion case: we shall sho w that there exists exactly one quaternion octic CM-field with class n umber 2, namely: Q(root alpha) with alpha=-(2+root 2)(3+root 3). Moreo ver, we shall show that the Hilbert class field of this number field i s a normal and non-abelian CM-field of degree 16 with class number one .