The finitude of tamely ramified pro-p extensions of number fields with cyclic p-class groups

Let p be an odd prime and F be a number field whose p-class group is cyclic. Let F_\ be the maximal pro-p extension of F which is unramified outside a single non-p-adic prime ideal q of F. In this work, we study the finitude of the Galois group G_\(F) of F_\ over F. We prove that G_\(F) is finite for the majority of q's such that the generator rank of G_\(F) is two, provided that for $p = 3$, F is not a complex quartic field containing the primitive third roots of unity.