Given a perfect field $F$ of prime characteristic the ring of Witt Vectors $W(F)$ is a discrete valuation ring. For example, $W(\mathbb{F}\_p)$ is the ring of $p$-adic integers. Is it possible to embed an arbitrary unramified discrete valuation ring of mixed characteristic into a Witt Vector ring of a perfect field?



discrete valuation ring and ring of witt vectors
I think this is possible, if I understand your question correctly. If $R$ is a discrete valuation ring of mixed characteristic $(0,p)$ with residue field $k$ and maximal ideal $pR$, then $R$ is a Cohen ring for $k$. Cohen rings are unique up to (generally non-unique) isomorphism, and there is a construction of the Cohen ring for any field $k$ of characteristic $p$ which realizes the ring as a subring of $W(k)$. Now $W(k)$ is a subring of $W(k^{p^{-\infty}})$, where $k^{p^{-\infty}}$ is the perfect closure of $k$. 


