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.

References in corpus (3)

• On definitions of polynomials over function fields of positive characteristi
• Hilbert's Tenth Problem over Function Fields of Positive Characteristic Not Containing the Algebraic Closure of a Finite Field
• Defining Subrings in Finitely Generated Fields of All Characteristics

Cited by in corpus (3)

• Bi-interpretability with $\mathbb{Z}$ and models of the complete elementary theories of ${\text{SL}}_n(\mathcal{O})$, ${\text{T}}_n(\mathcal{O})$ and ${\text{GL}}_n(\mathcal{O})$, $n\ge 3$
• Characterizing finitely generated fields by a single field axiom
• Defining Subrings in Finitely Generated Fields of All Characteristics