Maybe, I am being stupid, but when I consider ramified extension of a perfectoid field with the characteristic $0$, I cannot find the correspondent field with characteristic $p$. Let me put it more precisely.

Let $K$ be the completed field of $\mathbb{Q}_p(p^{-\infty})$. Consider $\zeta_p$ be a $p$th root of unity, and let $L=K(\zeta_p)$. According to Scholze's theorem (or Fontaine-Winterberger maybe enough), there exists a correspondent field $L^\flat$ which is of degree $p-1$ over $K^{\flat}$. However, I cannot find such $L^\flat$.

I tried to follow the definition to construct $L^\flat$ which is

$$L^\flat=\varprojlim L^{\circ}/p$$

where the transition map is Forbenius, and then I meet the problem that I cannot find the $p$th root of $\zeta_p$ in $L$. Similar problem happens if I consider $K$ to be the completion of $\mathbb{Q}_p(\zeta_{p^{\infty}})$ and adjoint $p$th root of $p$. However, the extension of a perfectoid field is a perfectoid field according to Scholze, so I do not think this should happen.

I would like to thanks all of you for your help.