Construction of the 2-Hilbert class field tower of certain biquadratic fields

Let p and q be prime numbers such that p = 1 mod 8, q = -1 mod 4 and (p/q) = 1, d = pq, k = Q(rootd, i), k(2)((1)) be the 2-Hilbert class field of k, k(2)((2)) be the 2-Hilbert class field of k(2)((1)) and G(2) be the Galois group of k(2)((2))/k. The 2-part C-k,C-2 of the class group of k is of type (2, 2), so k(2)((1)) contains three extensions K-i/k, i = 1, 2, 3. Our goal is to determine the group C-k,C-2, to study the problem of capitulation of the classes of k in K-i, i = 1, 2, 3 and to construct the 2-class field tower of k.