ON UNRAMIFIED GALOIS 2-GROUPS OVER Z2-EXTENSIONS OF SOME IMAGINARYBIQUADRATIC NUMBER FIELDS

For an imaginary biquadratic number field K = Q(root-q, root d), where q > 3 is a prime congruent to 3 (mod 8), and d is an odd square-free integer which is not equal to q, let K(infinity )be the cyclotomic Z(2)-extension of K. For any integer n >= 0, we denote by K(n )the nth layer of K-infinity/K. We investigate the rank of the 2-class group of K-n, then we draw the list of all number fields K such that the Galois group of the maximal unramified pro-2-extension over their cyclotomic Z(2)-extension is metacyclic pro-2 group.