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

Abstract

In 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. Academic Press

Maximal unramified extensions of imaginary quadratic number fields of small conductors, II

“…We know that there exist only finitely many nonnormal quartic CM-fields with relative class number two [12] and all such fields has been determined by H.-S. Yang and S.-H. Kwon [17]. We also note that in this case by [3, (iv), (a)] the 2-class field tower of K has length two and IK) C--Q2n (n >_ 4).…”