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Let $K$ be a mixed characteristic complete discrete valuation field withresidue field admitting a finite $p$-basis, and let $G_K$ be the Galois group.We first classify semi-stable representations of $G_K$ by weakly admissiblefiltered $(\varphi,N)$-modules with connections. We then construct a fullyfaithful functor from the category of \emph{integral} semi-stablerepresentations of $G_K$ to the category of Breuil-Kisin $G_K$-modules. Usingthe integral theory, we classify $p$-divisible groups over the ring of integersof $K$ by minuscule Breuil-Kisin modules with connections.

Integral $p$-adic Hodge theory in the imperfect residue field case