A criterion for perfectoid fields

TL;DR

This paper establishes a new criterion for identifying perfectoid fields among nonarchimedean fields using the tilting construction, linking algebraic properties of the tilt to perfectoidness.

Contribution

It introduces a novel characterization of perfectoid fields via their tilts, providing a practical criterion for detection among nonarchimedean fields.

Findings

A complete subfield of _p is perfectoid iff its tilt is not algebraic over _p.

The tilting construction can be used to detect perfectoid fields.

Includes conjectures relating APF extensions and 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.