Defining Subrings in Finitely Generated Fields of All Characteristics

Abstract

We give a construction of a large first-order definable family of subrings of finitely generated fields $K$ of any characteristic. We deduce that for any such $K$ there exists a first-order sentence ${\phi }_K$ characterising $K$ in the class of finitely generated fields, i.e. such that for any finitely generated field $L$ we have $L\models {\phi }_K$ if and only if $L\cong K$. This answers a question considered by Pop and others. In characteristic two, our results depend on resolution of singularities, whereas they are unconditional in all other characteristics.

Updated to include the case of characteristic two, conditional on resolution of singularities