A remark on integral representations associated with \(p\)-adic field extensions (Q1917588)

Let \(K\) be a local field with algebraically closed residue field of characteristic \(p>0\). Let \(K_\infty/K\) be a \(\mathbb{Z}_p\)-extension and \(K_m/K\) be its subextension of degree \(p^m\). For a product \(F\) of local fields denote by \(O(F)\) the product of the ring of integers of the factors. The main theorem of the paper is an extension of a result of \textit{S. Sen} [Invent. Math. 94, 1-12 (1988; Zbl 0695.12009)] and its generalization by \textit{F. Destrempes} [Acta Arith. 63, 267-286 (1993; Zbl 0777.11047)]. It states that if for two separable finite extensions \(E/K\) and \(E'/K\) the \(O(K_m)\)-semilinear representations of \(G(K_\infty/K_m)\) on the additive group of \(O(E \otimes_K K_m)\) and of \(O(E' \otimes_K K_m)\) are isomorphic, then \(E/K\) and \(E'/K\) are of the same degree and their Galois closures coincide.