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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 p1 ≡ p2 ≡-q ≡ 1(mod 4) be different prime integers. Put K = Q(√p1 p2 q), and denote by C,2 its 2-class group and by 2(1) (respectively 2(2)) its first (respectively second) Hilbert 2-class field. Then, we are interested in studying the metacyclicity of G = Gal( 2(2)/) and the cyclicity of Gal( 2(2)/ 2(1)) whenever the 4-rank of C,2 is 1.

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... Denote by H i (i = 1, 2, 3), the three maximal subgroups of G, then G is metacyclic if and only if d(H i ) ≤ 2 for each i = 1, 2, 3. Proof. If G is abelian the result follows easily, else see Proposition 2.1 in [3]. ...
... Proof. For the case δ = 1 see [3] and for the case δ = 2, the result is obtained proceeding similarly as in [3]. ...
... Proof. For the case δ = 1 see [3] and for the case δ = 2, the result is obtained proceeding similarly as in [3]. ...
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