# discrete valuation ring and ring of witt vectors

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?

• Any witt ring is unramified over Z_p, so I would guess no. – Dima Sustretov Jun 6 '15 at 14:07
• @Dima Do you, by ramified, mean that the residual characteristic $P$ lies in the square power of the maximal ideal of the discrete valuation ring? If so, no problem, I should modify the question as: Is it possible to embed an arbitrary unramified discrete valuation ring into a Witt Vector ring of a perfect field? – Aurora Jun 6 '15 at 14:17
• Notation: In the context of Witt vectors, the field with $p$ elements should definitely not be denoted by $\mathbb Z_p$! – ACL Jun 6 '15 at 17:01
• @ACL, the "error" has been fixed! – KConrad Jun 6 '15 at 18:24

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 $k$.