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