Let $K$ be a perfectoid field, i.e. a complete nonarchimedean field $K$ with non-discrete rank 1 value groups and characteristic $p$ residue field, such that the p-th power frobenius map $O_K/p \to O_K/p$ is surjective.
A paper I am reading says one can always find $\varpi \in O_K$ with $|p|\leq |\varpi|<1$ and $p$th-root $\varpi^{1/p} \in O_K$. Why? I think it should follow easily the fact the value group is not discrete, but I'm being stupid.
This follows from Lemma 3.2 in Scholze's Perfectoid Spaces paper, which says the value group of a perfectoid field is $p$-divisible.
Here is how the proof goes. Since the valuation is not discrete, the value group is not $|p|^\mathbb{Z}$, so there is $x \in K$ such that $|p|<|x|<1=|p|^0$. Then since Frobenius $O_K/p \to O_K/p$ is surjective, there exists $y \in O_K$, such that $y^p-x \in pO_K$, i.e. $|y^p -x|\leq |p|$. Then $|y^p| \leq \max\{|y^p-x|, |x|\}=|p|$ and we have $|y|^p=|p|$. Set $\varpi^{1/p}=y$.