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 $\varphi_K$ characterising $K$ in the class of finitely generated fields, i.e. such that for any finitely generated field $L$ we have $L \models \varphi_K$ 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.