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