  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.
