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