Imaginary quadratic fields with \(Cl_2(k)\simeq (2,2,2)\).
A characterization of imaginary quadratic fields, \(k\), with elementary 2-class group of rank 3 for which the Hilbert 2-class field has 2-class group of rank 2 is given. Assume the discriminant, \(d\), of \(k\) can be written as \(d= d_1d_2d_3d_4\) where \(d_1\), \(d_2\) and \(d_3\) are negative prime discriminants. Let \(k^1\) (respectively \(k^2\)) denote the Hilbert 2-class field of \(k\) (respectively \(k^1\)). Let \(G\) denote the Galois group of \(k^2/k\), set \(G_3= [G,G]\) and assume that \(G/G_3\) has order 32. 1. Among the unramified Abelian extensions of degree 16 over \(k(\sqrt{d_4})\), there is a unique extension \(K\) for which the Hilbert 2-class group \(Cl_2(K)\) of \(K\) has maximal rank; moreover, \(K/k\) is normal and \(\text{Gal}(K/k)\cong G/G_3\). 2. The following statements are equivalent: (i) \(k^2\neq k^3\); (ii) \(\text{rank\,}Cl_2(k^1)= 3\); (iii) \(G/G_3\cong 32.033\); (iv) \(d= d_1d_2d_3d_4\), where \(d_1,d_2,d_3< 0\) and \(d_4> 0\) are prime discriminants such that \((d_1/p_2)= (d_2/p_3)= (d_3/p_1)= (d_1/p_4)= -1\) and \((d_4/p_2)= (d_4/p_3)= 1\), where \(p_i\) is the unique prime dividing \(d_i\). If \(k^2= k^3\) then rank \(Cl_2(k^1)= 2\).
scientific article; zbMATH DE number 2012762