Associate a discrete valuation ring to a field $k$.

Question

I have a field $k$ of positive characteristic $p$, not necessarily perfect. Can i find a discrete valuation ring that have $k$ as residue field and field of fractions $K$ of characteristic zero?

I see the Witt construction for perfect field and its properties, but I don't find anything for a general case (or more probably I didn't understand well this construction).

Answer 1

What you're looking for is a Cohen ring.

The proof that Cohen rings exist can be found here:

http://stacks.math.columbia.edu/tag/0323