Determination of all Quaternion Octic CM-Fields with Class Number 2

Abstract

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 CMfield 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(V<x) wi tha = ( 2 + y'2)(3 + V3). 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. 1. Factorization of the Dedekind zeta function of a quaternion octic number field Let N be a quaternion octic CM-field, that is, N/Q is a normal octic extension with Galois group Gal (N/Q) the quaternion group Q8 with eight elements Q8 = {± 1, ± U ±j, ± k} with i = / = k = — 1 and ij = —ji = kjk = — kj = i and hi = — ik = j . Let N be the quartic subfield of N, so that N is the maximal totally real subfield of N. Let ki5 k and kfc be the three quadratic subfields of N, so that these quadratic fields are real quadratic fields. The quartic extensions N/k4, N/k; and N/kfc are cyclic and the quartic extension N/Q is biquadratic bicyclic. The lattice of subfields can be set out as follows: