Extra capitulation and central extensions (Q1345278)

Let \(L/K\) be an unramified cyclic extension of algebraic number fields of finite degree. Let \(C_K\) denote the ideal class group of \(K\). An ideal class of \(C_K\) is said to capitulate in \(L\) if it becomes principal in \(C_L\), and Hilbert's Theorem 94 assures that at least \([L : K]\) ideal classes capitulate in \(L\). It is said that extra capitulation occurs if more than \([L : K]\) ideal classes capitulate. The paper is concerned with the relation between extra capitulation and the existence of a central extension of \(L^{(1)} M/K\) with \(M\) some abelian extension of \(K\), where \(L^{(1)}\) denotes the Hilbert class field of \(L\). \textit{K. Iwasawa} [J. Math. Pures Appl., IX. Sér. 35, 189-192 (1956; Zbl 0071.265)] proved that the subgroup of \(C_K\) which capitulates in \(C_L\) is isomorphic to \(H^1 (G, E_L)\), where \(G = \text{Gal} (L/K)\) and \(E_L\) denotes the group of global units of \(L\). Using this fact and the property of the Herbrand quotient the author first proves that extra capitulation occurs if and only if \(H^1 (G, U_L/E_L) \neq \{1\}\), where \(U_L\) is the unit ideles of \(L\). Starting with this result the author studies the realization of \(H^1 (G,U_L/E_L)\) as a Galois group of finite abelian extensions of \(L\). The main result of the paper is that \(H^1 (G,U_L/E_L)\) is isomorphic to \(\text{Gal} (L'/K^{ab} L^{(1)})\), where \(K^{ab}\) and \(L'\) denote the maximal abelian extension of \(K\) and the maximal central extension of \(L^{(1)} K^{ab}/K\) contained in \(L^{ab}\), respectively. Also proved is the existence finite abelian extensions \(F_1/F_2\) of \(L\) such that \(\text{Gal} (L'/K^{ab} L^{(1)})\) is isomorphic to the subgroup of \(\text{Gal} (F_1/F_2)\) which is generated by the inertia groups of \(\text{Gal} (F_1/L)\) contained in \(\text{Gal} (F_1/F_2)\).