# Is the completion of the field generated by torsion points of a 1-dimensional formal group perfectoid?

Let $$K$$ be a finite extension of $$\mathbb{Q}_p$$ and let $$G$$ be a 1-dimensional formal group defined over $$\mathcal{O}_K$$. Consider the field $$K_\infty$$ obtained by adjoining to $$K$$ all the solutions to $$[p]^n(x) = 0$$ with $$x \in \mathbb{C}_p$$ with $$|x|<1$$ and $$n\geq1$$. (These torsion points depend on a choice of coordinate for the formal group, but the field $$K_\infty$$ does not). Equivalently, if $$V(G)$$ is the Tate module of $$G$$, considered as a $$\mathrm{Gal}(\overline{K}/K)$$-representation, the field $$K_\infty$$ is the subfield of $$\overline{K}$$ which corresponds to its kernel via Galois theory.

Consider the $$p$$-adic completion $$\widehat{K}_\infty$$ of $$K_\infty$$. My Question is: is $$\widehat{K}_\infty$$ perfectoid?

At the very least, I know that $$K_\infty$$ must be infinitely ramified (the Tate module of $$G$$ has a nonzero Hodge-Tate weight). Also, the answer is yes when $$G$$ is a Lubin-Tate formal group.

Thanks!

Yes it is. The extension $$K_\infty/K$$ is deeply ramified. Now apply proposition 6.6.6 of Gabber-Ramero's "Almost ring theory".