Characterizing finitely generated fields by a single field axiom

TL;DR

This paper proves that finitely generated fields can be uniquely characterized by a single first-order sentence, resolving a longstanding problem in field theory, with some results depending on unresolved conjectures.

Contribution

It establishes a first-order axiomatization for finitely generated fields, linking elementary equivalence to isomorphism, conditional on singularity resolution in characteristic two.

Findings

Unique first-order characterization for finitely generated fields

Conditional proof depending on resolution of singularities in characteristic two

Unconditional results for all other characteristics

Abstract

We resolve the strong Elementary Equivalence versus Isomorphism Problem for finitely generated fields. That is, we show that for every field in this class there is a first-order sentence which characterizes this field within the class up to isomorphism. Our solution is conditional on resolution of singularities in characteristic two and unconditional in all other characteristics.