Explicit example of complete discrete valuation field with prescribed residue field

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 $$X\to\operatorname{Spec}\mathbb Z$$

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 $$k[\![t]\!]$$