  Let K be a p-adic local field with residue field k such that
[k:k^p]=p^e<\infty and V be a p-adic representation of Gal(\bar{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(\bar{K}/K) if and only if V is a potentially crystalline (resp. potentially
semi-stable) representation of Gal(\bar{K^{pf}}/K^{pf}) where K^{pf}/K is a
certain p-adic local field whose residue field is the smallest perfect field
k^{pf} containing k. As an application, we prove the p-adic monodromy theorem
of Fontaine in the imperfect residue field case.
