Value groups of Henselian valuations (Q1318929)

A system \((G, A,\deg, v)\) is called admissible situation over \(k\), if \(G\) is a pro-finite group with \((G_ K)\) closed subgroups of \(G\) indexed by fields \(K\), \(G= G_ k\), \(A\) is a multiplicative \(G\)-module, \(\deg: G\to \widehat {\mathbb{Z}}\) a surjective continuous homomorphism and \(v_ K: A_ K\to \widehat {\mathbb{Z}}\) is a Henselian valuation with respect to deg, where \(A_ K\) is the submodule of \(A\) of elements fixed by \(G_ K\). The authors then prove that if \((G,A, \deg,v)\) and \((G,A, \deg,v')\) are admissible situations such that the class field axiom holds for any subextension \(L/K\) of \(K\widetilde{k}/K\) in both situations (where \(\widetilde{k}\) is such that \(G_{\widetilde{k}}\) is the kernel of deg), the valuations \(v_ K\) and \(v'_ K\) have the same unit group.