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

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.