On a power integral bases problem over cyclotomic \(\mathbb{Z}_p\)-extensions (Q1841818)

Let \(k\) be a complex abelian extension containing a primitive \(p\)-th root of unity \(\zeta_p\), and let \(k^+\) be its maximal real subfield. Also, let \(k_n\) denote the \(n\)-th layer of the cyclotomic \(\mathbb Z_p\)-extension of \(k\). The author shows that if Greenberg's conjecture holds for \(k^+\) (that is, if \(\lambda(k^+) = 0\)), then every unramified cyclic extension \(L/k_n\) of degree \(n\) that is normal over \(k^+\) with a Galois group isomorphic to a generalized dihedral group has a power integral basis.