On the Hilbert 2-class field of some quadratic number fields

In this paper, we investigate the cyclicity of the 2-class group of the first Hilbert 2-class field of some quadratic number field whose discriminant is not a sum of two squares. For this, let p(1) p(2) -q 1 (mod 4) be different prime integers. Put k = Q(root p(1)p(2)q), and denote by C-k,C-2 its 2-class group and by k(2)((1)) (respectively k(2)((2))) its first (respectively second) Hilbert 2-class field. Then, we are interested in studying the metacyclicity of G = Gal(k(2)((2))/k) and the cyclicity of Gal(k(2)((2))/k(2)((1))) whenever the 4-rank of C-k,C-2 is 1.