Éz fields

Abstract

Let $K$ be a field. The étale open topology on the $K$-points $V(K)$ of a $K$-variety $V$ was introduced in our previous work. The étale open topology is non-discrete if and only if $K$ is large. If $K$ is separably, real, $p$-adically closed then the étale open topology agrees with the Zariski, order, valuation topology, respectively. We show that existentially definable sets in perfect large fields behave well with respect to this topology: such sets are finite unions of étale open subsets of Zariski closed sets. This implies that existentially definable sets in arbitrary perfect large fields enjoy some of the well-known topological properties of definable sets in algebraically, real, and $p$-adically closed fields. We introduce and study the class of éz fields: $K$ is éz if $K$ is large and every definable set is a finite union of étale open subsets of Zariski closed sets. This should be seen as a generalized notion of model completeness for large fields. Algebraically closed, real closed, $p$-adically closed, and bounded $\mathrm{PAC}$ fields are éz. (In particular pseudofinite fields and infinite algebraic extensions of finite fields are éz.) We develop the basics of a theory of definable sets in éz fields. This gives a uniform approach to the theory of definable sets across all characteristic zero local fields and a new topological theory of definable sets in bounded $\mathrm{PAC}$ fields. We also show that some prominent examples of possibly non-model complete model-theoretically tame fields (characteristic zero Henselian fields and Frobenius fields) are éz.