Crystalline and semi-stable representations in the imperfect residue field case

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Let K be a p-adic local field with residue field k such that [k:kᵖ]=pᵉ<∞ and V be a p-adic representation of Gal(K/K). Then, by using the theory of p-adic differential modules, we show that V is a potentially crystalline (resp. potentially semi-stable) representation of Gal(K/K) if and only if V is a potentially crystalline (resp. potentially semi-stable) representation of Gal(Kpf/Kpf) where Kpf/K is a certain p-adic local field whose residue field is the smallest perfect field kpf containing k. As an application, we prove the p-adic monodromy theorem of Fontaine in the imperfect residue field case.