Ramification invariants and torsion Galois module structure in number fields
Let K/k be a normal extension of number fields with Galois group \(\Gamma\) and rings of integers \({\mathfrak O}\) resp. \({\mathfrak o}\). By the work of Fröhlich, Taylor and others much information has been obtained about the \({\mathbb{Z}}\Gamma\)-module (resp. \({\mathfrak o}\Gamma\)-module) \({\mathfrak O}\) and other \(\Gamma\)-modules related to K. In the article under review two further Galois modules are investigated which naturally intervene in the previous work. One is the \({\mathfrak o}\)-torsion module T(K/k) :\(=\) different (K/k)\({}^{-1}/{\mathfrak O}\). If K/k is tame an explicit structure theorem for this \({\mathfrak o}\Gamma\)-module is shown, which among other things allows to extend Taylor's result, that \({\mathfrak O}\) is stably self-dual over \({\mathbb{Z}}\Gamma\). The second torsion Galois module R(K/k) under consideration is defined as the cokernel of the natural injection \(\Psi\) : \({\mathfrak O}\otimes_{{\mathfrak o}}{\mathfrak O}\to {\mathfrak O}\Gamma\), \(x\otimes y\mapsto \sum_{\sigma \in \Gamma}x.\sigma (y)\sigma.\) It is shown that the \({\mathfrak O}\Gamma\)-module R(K/k) determines completely the local Galois module structure of \({\mathfrak O}\). T(K/k) and R(K/k) are related by a short exact sequence which generalizes the fact that the discriminant \(\delta\) (K/k) is the square of an ideal of \({\mathfrak O}\). The main result on R(K/k) is the complete determination of its primary components for all primes at which K/k is tame. Here various Stickelberger elements enter into the theory. Conversely, by a similar approach as in [\textit{A. Fröhlich}, Algebr. number fields, Proc. Symp. Lond. Math. Soc., Univ. Durham 1975, 589-607 (1977; Zbl 0376.12002)] as a corollary a weak form of the classical Stickelberger theorem is obtained. In general one expects that T(K/k) and R(K/k) contain relevant information concerning the \(\Gamma\)-module structure of \({\mathfrak O}\) which becomes trivial for unramified extensions. The results of this article seem to support this point of view.
scientific article; zbMATH DE number 3908543
scientific article