Pseudo T-closed fields

TL;DR

This paper introduces a unified framework for studying various pseudo-closed fields through pseudo T-closed fields, revealing their model-theoretic properties and classifications, especially for bounded cases.

Contribution

It proposes a general theory encompassing pseudo algebraically closed, pseudo real closed, and pseudo p-adically closed fields, and provides classification results under certain conditions.

Findings

Pseudo T-closed fields satisfy a local-global principle.

Bounded pseudo T-closed fields are NTP2 with finite burden.

The framework describes fields with multiple V-topologies.

Abstract

Pseudo algebraically closed, pseudo real closed, and pseudo $p$-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately. In this text, we propose a unified framework for studying them: the class of pseudo $T$-closed fields, where $T$ is an enriched theory of fields. These fields verify a "local-global" principle for the existence of points on varieties with respect to models of $T$. This approach also enables a good description of some fields equipped with multiple $V$-topologies, particularly pseudo algebraically closed fields with a finite number of valuations. One important result is a (model theoretic) classification result for bounded pseudo $T$-closed fields, in particular we show that under specific hypotheses on $T$, these fields are NTP$_2$ of finite burden.