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Let's fix the field $\overline F:=\mathbb F_p(t)$. What is the explicit expression of a complete discrete valuation field $F$ of characteristic $0$ which has $\overline F$ as residue field?

Note that I'm looking for characteristic $0$, because the characteristic $p$ case is easy. It is enough to take $\mathbb F_p(t)((u))$.

This kind of field has a precise geometric meaning, indeed is the complete discrete valuation field corresponding to vertical curves on an arithmetic surface $X\to\operatorname{Spec}\mathbb Z$

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In general, if $k$ is a perfect field, there is an unique complete discrete valuation ring with residue field k and quotient field of characteristic 0. This is the ring of Witt vectors over $k$ and it is denoted as $W(k)$. This is called a DVR of mixed characteristic.

The details of the construction of $W(k)$ can be found in Lang's Algebra or in Serre's Local Fields; the latter should also contain a proof of the above statement.

Interestingly enough, there is also a similar statement for equal characteristic DVR's: there is only an unique complete DVR with prescribed perfect residue field $k$, and this is $k[\![t]\!]$

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