Largeness and generalized t-henselianity

TL;DR

This paper characterizes large fields in terms of a new class of field topologies called generalized t-henselian topologies, linking algebraic largeness with topological properties where the implicit function theorem applies.

Contribution

It establishes an equivalence between largeness of countable fields and the existence of generalized t-henselian topologies, and characterizes the étale open topology via these topologies.

Findings

Large fields admit gt-henselian topologies.

The étale open topology is the intersection of all gt-henselian topologies.

Implicit function theorem holds in gt-henselian topologies.

Abstract

Let $K$ be a countable field. Then $K$ is large in the sense of Pop if and only if it admits a field topology which is "generalized t-henselian" (gt-henselian) in the sense of Dittmann, Walsberg, and Ye, meaning that the implicit function theorem holds for polynomials. Moreover, the \'etale open topology can be characterized in terms of the gt-henselian topologies on $K$: a subset $U \subseteq K^n$ is open in the \'etale open topology if and only if it is open with respect to every gt-henselian topology on $K$.