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We extend Schoof's theorem from cyclic case to any finite case and apply this to construct a class of Z/m Z Z/ (n) Z extensions of Q, where m is either a power of 2 or an odd integer, and n be any integer such that m divides n. As an application, we give some number fields with small root discriminant, having infinite p-class field tower when p=3, 5, 7.
Xing et al. (Sun,) studied this question.