local field extension

Question

let $k$ a perfect field and consider the following field $F= k((\omega))$, complete w.r.t a valuation for which $\omega$ is a uniformizer. Consider the field extension $E=F[T]/(T^2-\omega^3)$.

1. Is it true that $\mathcal{O}_F= k[[\omega]]$ ?
2. What is it $\mathcal{O}_E$?
3. let $x\in E$ a root of $T^2-\omega^3$. How far are $\mathcal{O}_E$ and $\mathcal{O}_F[x]$?

Answer 1

Hint: The polynomial $T^2- \omega$ is Eisenstien polynomial over $\mathcal{O}_F$ for which $E$ is the splitting field over $F.$