Is Whaples' theorem a group theoretical result? (Q1315105)

A theorem of \textit{G. Whaples} [Duke Math. J. 24, 201-204 (1957; Zbl 0081.267)] states, that if a field admits a cyclic extension of degree \(p\), where \(p\) is an odd prime, it admits a pro-cyclic extension of degree \(p^ \infty\). Similarly, the existence of a cyclic extension of degree 4 implies the existence of a pro-cyclic extension of degree \(2^ \infty\). The author shows that Whaples' result does not generalizes to torsion free pro-finite groups. Namely, for an odd prime \(p\) \((p=2)\) and a natural number \(n\) (respectively, \(n\geq 3\)) he gives examples of torsion free pro-\(p\)-groups with \(Z/ p^ n Z\), but not \(Z/ p^{n+1} Z\), as a factor.