Perfectoid fields in the language of rings

TL;DR

This paper characterizes the class of all perfectoid fields with a fixed residue characteristic p using ring language, extending previous work by the authors and Kartas.

Contribution

It provides a new algebraic description of perfectoid fields in the language of rings, building on prior foundational work.

Findings

Identifies the elementary class generated by perfectoid fields in ring language.

Extends previous characterizations of perfectoid fields.

Provides a basis for further algebraic and model-theoretic analysis.

Abstract

Building on work of the first author and Kartas, we identify the elementary class generated by all perfectoid fields of fixed residue characteristic $p$ in the language of rings.