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_{q} 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_{q}(F) of F_{q} over F. We prove that G_{q}(F) is finite for the majority of q's such that the generator rank of G_{q}(F) is two, provided that for p=3, F is not a complex quartic field containing the primitive third roots of unity.