Cyclotomic units in \(\mathbb{Z}_p\)-extensions (Q1196307)

Let \(K_ n=\mathbb{Q}(\zeta_{p^{n+1}}+ \zeta_{p^{n+1}}^{-1})\), where \(p\) is an odd prime and \(\zeta_{p^k}\) is a primitive \(p^k\)-th root of unity. Let \(A_n\) denote the \(p\)-part of the ideal class group of \(K_n\), and \(B_n\) the \(p\)-part of the factor group of the cyclotomic units in the unit group of \(K_n\). The authors describe the structure of the direct and inverse limits of \(B_n\). In particular, \(\varinjlim B_n\simeq (\mathbb{Q}_p/\mathbb{Z}_p)^\lambda \oplus M\), where \(\lambda\) is the Iwasawa invariant and \(M\) is a finite group. It is proved that \(M\simeq C\) where \(C\) is determined by \(\varprojlim A_n\simeq \mathbb{Z}_p^\lambda \oplus C\).