On the structure of certain valued fields

In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete fields K 1 and 2 with perfect residue fields, show that if n -th rings are isomorphic for each ? , then isometric isomorphic. More generally, there is depending only on ramification indices such homomorphism from ring to can be lifted a between valuation rings. Moreover, get functor category certain principal Artinian local length which naturally generalizes functorial property unramified Our lifting result improves Basarab's relative completeness theorem henselian solves question posed by Basarab, in case