LOWER BOUNDS FOR RELATIVE CLASS-NUMBERS OF CM-FIELDS

Let K be a CM-field that is a quadratic extension of a totally real nu mber field k. Under a technical assumption, we show that the relative class number of K is large compared with the absolute value of the dis criminant of K, provided that the Dedekind zeta function of k has a re al zero s such that 0 < s < 1 . This result will enable us to get shar p upper bounds on conductors of totally imaginary abelian number field s with class number one or with prescribed ideal class groups.