# Discrete valuation rings.

Given an algebraically closed field $\mathbb F$ of characteristic $p$, let $\mathbb A$ be a discrete valuation ring of characteristic zero having $\mathbb F$ as its residue field ( it does exist, but I don't know a reference).

(1) If $\zeta$ is a primitive cubic root of unity, then $\mathbb A[\zeta]$ is also a discrete valuation ring having $\mathbb F$ as residue field? Moreover, if the characteristic of $\mathbb F$ is not 3, then $\zeta\in\mathbb A$? (What about for a $m^{th}$ root of unity?)

(2) If the characteristic of $\mathbb F$ is not 2, then $\sqrt2\in\mathbb A$?

What could be a good reference to attack these questions?

Well, I've found that the answer for (1) and (2) is 'YES'. But it is direct consequence of Henselian Fields. So, the question now becomes the following:

Given an algebraically closed field $\mathbb F$ of characteristic $p$, how to prove that there exists a Henselian discrete valuation ring of characteristic zero having $\mathbb F$ as its residue field ???

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About the reference: Google "Witt vectors" :) –  Wanderer Oct 18 '11 at 1:09
And for your questions: you could try to apply Hensel's lemma? –  Wanderer Oct 18 '11 at 1:13
Sometimes I think people apply the "research level" mantra a little too often. MO is very largely for mathematicians who are seeking answers to questions in areas where they are not expert. As Gerry Myerson once said, when it comes to areas outside our domain, we are all at the graduate student level, and I think that reminder might apply here. –  Todd Trimble Oct 18 '11 at 17:18
For any perfect field $\mathbb F$ of characteristic $p$, so in particular for any algebraically closed field, the Witt vectors $W(\mathbb F)$ are a complete, hence Henselian, discrete valuation ring of characteristic 0 having $\mathbb F$ as residue field. –  user2035 Oct 19 '11 at 6:09
Serre's "Corps Locaux" (or Local Fields) is a great reference to look up Witt vectors! –  Tommaso Centeleghe Oct 19 '11 at 14:24