Unramified abelian extensions of number fields (Q1105643)

Let \(F'\) denote the Hilbert class field of a number field \(F\) and let \(F\subseteq K\subseteq F'\); let \(G\) be the Galois group of the extension \(K\mid F\). The author obtains some necessary conditions for the class- group \(Cl_K\) (or \(Cl_{F'})\) to be cyclic. If \(G\) is an abelian \(p\)-group of the type \((p^{n_1},p^{n_2},\ldots,p^{n_r})\) with \(n_1\ge n_2\ge\ldots\ge n_r\), \(n_2\ge 1\), it is proved that \(p^{n_1+1}\) divides \(| \operatorname{Ker}(j)|\) where \(j\colon Cl_F\to Cl_K\) is induced by lifting of ideals from \(F\) to \(K\). In continuation of his previous work [J. Number Theory 13, 246--254 (1981; Zbl 0458.12007)], the author studies the epimorphism \(\lambda\colon H^{-1}(G,Cl_K)\to \operatorname{Ker}(j)\cap N_{K/F}(Cl_K)\) \((H^{-1}(G,Cl_K)\) means the Tate cohomology group) and proves that \(\operatorname{Ker}(\lambda)\) is isomorphic to the Galois group of the central class field of \(K\) (with respect to \(F)\) over \(F'\); it follows then from a theorem of \textit{K. Miyake} [Nagoya Math. J. 96, 83--94 (1984; Zbl 0577.12008)] that \(| G|\) divides \(| \operatorname{Ker}(j)|\) when \(\lambda\) is an isomorphism. Finally the \(p\)-rank of \(\operatorname{Ker}(\lambda)\) is investigated and a necessary condition for the \(p\)-class tower of \(F\) to be of length one is given.