The class number one problem for some non-abelian normal CM-fields

Let N be a non-abelian normal CM-field of degree 4p, p any odd prime. Note that the Galois group of N is either the dicyclic group of order 4p, or the dihedral group of order 4p. We prove that the (relative) class number of a dicyclic CM-field of degree 4p is always greater then one. Then, Rie determine all the dihedral CM-fields of degree 12 with class number one: there are exactly nine such CM-fields.