The absolute Galois group \(G\) of a \(p\)-adic field \(K\) carries the natural filtration induced by the ramification subgroups. In this paper it is shown that, for finite extensions \(K_1\) and \(K_2\) over \(\mathbb{Q}_p\), any isomorphism of \(G_1\) and \(G_2\) respecting this ramification filtration yields a \(\mathbb{Q}_p\)-isomorphism \(\widetilde{\alpha}\) of \(K_1\) and \(K_2\); moreover, the canonical isomorphism of \(G_1\) and \(G_2\) resulting from \(\widetilde{\alpha}\) differs from \(\alpha\) by an inner automorphism. Without the filtration condition there is no such strong result [M. Jarden and J. Ritter, J. Number Theory 11, 1-13 (1979; Zbl 0403.12020)].
The proof starts out by showing that the topological group \(G\) together with its ramification filtration determines the ring \({\mathfrak O}_{\overline{K}}\) of integers and the \(p\)-adic completion \(\overline{K}^\wedge\) of the algebraic closure \(\overline{K}\) of \(K\). The main ingredient then is the observation that whether or not a finite-dimensional \(\mathbb{Q}_pG\)-module \(V\) is Hodge-Tate it is encoded in the filtered \(G\). The content of the paper is related to F. Pop [J. Reine Angew. Math. 392, 145-175 (1988; Zbl 0671.12005)] and J. Koenigsmann [J. Reine Angew. Math. 465, 165-182 (1995; Zbl 0824.12006)].