Let $k$ be a field of positive characteristic $p$. Is there necessarily a discrete valuation ring of characteristic $0$ with maximal ideal $(p)$ and residue field isomorphic to $k$?



Is every field the residue field of a discretely valued field of characteristic 0?
Yes, by Hasse-Schmidt ("Die Struktur diskret bewerteter Koerper", Crelle's Journal, 1934) for any field $k$ of characteristic $p$ there exists a strict Cohen ring $A$, which is a Noetherian, complete local, discrete valuation ring with maximal ideal $pA$ and residue field $A/pA = k$. See also Mac Lane (Theorem 2 in "Subfields and automorphism groups of p-adic fields", Ann. of Math. 1939) and Cohen ("On the structure and ideal theory of complete local rings", Trans. AMS, 1946).


