ON THE CLASS NUMBER ONE PROBLEM FOR NONNORMAL QUARTIC CM-FIELDS

We give explicit upper bounds for the discriminants of the non-normal quartic CM-fields with class number one, and for the discriminants of the dihedral octic CM-fields with class number one. These upper bounds are too large to enable us to achieve the determination of these number fields. Nevertheless, whenever a real quadratic number field k is fixed, we can explain how to determine the non-normal quartic CM-fields or the dihedral octic CM-fields with class number one and with real quadratic subfield k.