Let be an imaginary quadratic field with . It is known that the length of the Hilbert -class field tower is at least . Gerth (On 2-class field towers for quadratic number fields with -class group of type , Glasgow Math. J. 40(1) (1998), 63-69) calculated the density of where the length of the tower is ; that is, the maximal unramified -extension is a -extension. In this paper, we shall extend this result for generalized quaternion, dihedral, and semidihedral extensions of small degrees.
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