For any positive integer Formula: see text, we show that there exists a real number field Formula: see text (resp. Formula: see text) of degree Formula: see text whose Formula: see text-class group is isomorphic to Formula: see text such that the Galois group of the maximal unramified extension of Formula: see text (resp. Formula: see text) over Formula: see text (resp. Formula: see text) is abelian (resp. non abelian, more precisely isomorphic to Formula: see text or Formula: see text, the quaternion and the dihedral group of order Formula: see text respectively). In fact, we construct the first examples in the literature of families of real biquadratic fields for which the layers of the cyclotomic Formula: see text-extension satisfy the previous conditions and whose unramified abelian Formula: see text-Iwasawa modules are isomorphic to Formula: see text; hence these fields satisfy Greenberg’s conjecture.
Mohamed Mahmoud Chems-Eddin (Fri,) studied this question.