On the maximal unramified pro-2-extension of ℤ ₂ -extension of certain real biquadratic fields

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.