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 ???