On the local-global principle for embedding problems over global fields
Let \(K\) be a global field and \(G_k\) its absolute Galois group. Let \(A\) be a finite \(G_K\)-module. The action of \(G_K\) on \(A\) factors through a finite group \(G\), and we can consider embedding problems of the form \[ \begin{tikzcd} &&& G_K\ar[dl]\ar[d,"\varphi"] & \\ 1\ar[r] & A\ar[r] & E \ar[r] &G\ar[r] \ar[d] & 1\\&&& 1\end{tikzcd}\tag{\(\mathcal E\)} \] where \(\varphi\) is the canonical projection and \(E\) is a group extension of \(A\) with \(G\). The localization at a prime \({\mathfrak p}\) of \(K\) of such an embedding problem is: \[ \begin{tikzcd} &&& G_{K_{\mathfrak{p}}}\ar[dl]\ar[d] & \\ 1\ar[r] & A\ar[r] & E_{\mathfrak{p}} \ar[r] &G_{\mathfrak{p}}\ar[r] \ar[d] & 1\\&&& 1\end{tikzcd}\tag{\(\mathcal E_{\mathfrak{p}}\)} \] where \(G_{K_{{\mathfrak p}}}\leq G_K\) is the absolute Galois group of the completion \(K_{{\mathfrak p}}\), \(G_{{\mathfrak p}}= \varphi(G_{K_{{\mathfrak p}}})\) the decomposition group of \({\mathfrak p}\) in \(G\), and the \(E_{{\mathfrak p}}\) the inverse image of \(G_{{\mathfrak p}}\) under the map \(E\to G\). It is well known that the solvability of the global embedding problem \(({\mathcal E})\) implies the solvability of its localizations \(({\mathcal E}_{{\mathfrak p}})\) for each \({\mathfrak p}\), the reciprocal being not true in general. In the interesting paper under review, the author proves the following Theorem: For every global field \(K\) of characteristic \(\neq 2\) and any given \(m\geq 3\), there exists a \(G_K\)-module \(A\) of order \(2^m\) such that an embedding problem \(({\mathcal E})\) is locally solvable for each \({\mathfrak p}\) but not yet globally solvable. Modules \(A\) of the theorem are cyclic groups of order \(2^m\) with the action of \(G_K\) defined suitably. The construction of the proof is based on results of \textit{K. Hoechsmann's} paper [J. Reine Angew. Math. 229, 81--106 (1968; Zbl 0185.11202)].
scientific article; zbMATH DE number 2114659