Distinguishing every finitely generated field of characteristic \neq2 by a single field axiom

TL;DR

This paper proves that each finitely generated field of characteristic not 2 can be uniquely characterized by a single explicit field axiom within the language of fields, extending previous foundational results.

Contribution

It introduces a single explicit axiom that uniquely identifies the isomorphism type of any finitely generated field of characteristic not 2, advancing the field of model theory of fields.

Findings

Each such field's isomorphism type is encoded by one explicit axiom.

The result generalizes earlier work by Robinson, Rumely, Poonen, Scanlon, and the author.

The axiom precisely characterizes the field within the language of fields.

Abstract

We show that the isomorphy type of every finitely generated field $K$ with $\chr(K)\neq2$ is encoded by a \textit{\textbf{single\ha3explicit\ha3axiom}} $\istp K\!$ \textit{\textbf{in\ha3the\ha3language\ha3of\ha3fields}}, i.e., for all finitely generated fields $L$ one has: $\istp K$ holds in $L$ if and only if $K\cong L$ as fields. This extends earlier results by \nmnm{\footnotesize\sc Julia Robinson, Rumely, Poonen, Scanlon}, the author, and others.