If the characteristic of the coefficient field is the same as that of the residual characteristic of the local field, we need tools from -adic Hodge theory.
Cohen structure theorem: Suppose is equal characteristic CDVR (complete discrete valuation ring). Then the canonical ring homomorphism splits ring theoretically, i.e. there exists subring such that is an isomorphism, and any choice of induces . When is mixed characteristic , in this case, there is a coefficient subring with the following property: 1. is a CDVR with residue field isomorphic to via . 2. is absolutely unramified, i.e. unramified over , or . This is determined up to isomorphism by , but not uniquely/functorially. Such a coefficient ring is also called Cohen -ring.
If is not perfect, then can be very noncanonical. For example, if we take , and . Then is a CDVR with unique maximal ideal and . But is an automorphism of that reduced to identity mod .
If is perfect, there is a functor of Witt vectors from the category of perfect fields of characteristic to that of Cohen -rings, splitting the functor of taking reduction mod , and it is unique up to unique isomorphism. The functor uniquely extends to a functor from the category of perfect (Frobnius map is an isomorphism) -algebras to that of -adically complete () flat (torsion-free) -algebras with perfect.
Example: , .
-adic expansions: If , for some , then (expand ). There is a unique map preserving multiplication such that and has -th root for all .
The idea is to construct a Cauchy sequence that reduces to . Take such that (which exists since is perfect). Pick any lift , then . By completeness of , converges.
Since is flat, we can divide by and get a -adic expansion .
The question is that if we can write down using and . First . Then . Raising both sides to the -th power, then we get . So on and so forth.
For multiplication , we similarly expand and find out . In general, given , we can find such that is a polynomial over independent of .