Galois cohomology of a $p$-adic field via $(\mathrm{\Phi },\mathrm{\Gamma })$-modules in the imperfect residue field case

For a $p$-adic local field $K$ with perfect residue field, L. Herr constructed a complex which computes the Galois cohomology of a $p$-torsion representation of the absolute Galois group of $K$ by using the theory of $(\mathrm{\Phi },\mathrm{\Gamma })$-modules. We shall generalize his work to the imperfect residue field (the residue field has a finite $p$-basis) case.