A criterion for perfectoid fields

Abstract: The tilting correspondence is a fundamental property of perfectoid fields. In this note, we show that the tilting construction can also be used to detect perfectoid fields among nonarchimedean fields. In particular, for $K$ a complete subfield of $\mathbb{C}_p$ (a completed algebraic closure of $\mathbb{Q}_p$), $K$ is perfectoid if and only if its tilt is not algebraic over $\mathbb{F}_p$. We also include some conjectures on APF (arithmetically profinite) extensions, perfectoid fields, and their relations.