COMPUTATION OF RELATIVE CLASS-NUMBERS OF CM-FIELDS - APPLICATION TO OCTIC QUATERNION CM-FIELDS

We give an expression for the relative class number of any CM-field th at is suitable for numerical computation. This expression is then used in order to compute relative class numbers of octic CM-fields with Ga lois group the quaternion group of order eight. These computations hav e already been used in order to determine all octic quaternion CM-fiel ds with class number 2 (see [5]). Moreover, they will be used in a for thcoming paper [7] by the author and Ryotaro Okazaki in order to deter mine all octic quaternion CM-fields with ideal class groups of exponen t two.