The Non-normal Quartic CM-Fields and the Octic Dihedral CM-Fields with Relative Class Number Two

AbstractIn this paper we determine all non-normal quartic CM-fields with relative class number two and all octic dihedral CM-fields with relative class number two: there are exactly 254 non-isomorphic non-normal quartic CM-fields with relative class number two and 95 non-isomorphic octic dihedral CM-fields with relative class number two. This result permits us to find all octic dihedral CM-fields of which ideal class groups are nontrivial and cyclic of 2-power order: there are precisely 16 non-isomorphic such octic dihedral CM-fields