\'Ez fields

TL;DR

This paper introduces 'ez fields', a class of large fields where definable sets have well-behaved topological properties, unifying various theories of definable sets across different types of fields.

Contribution

The paper defines 'ez fields' as a new class of large fields with a generalized model completeness property, and develops their theory of definable sets, unifying multiple existing frameworks.

Findings

Existentially definable sets in perfect large fields are finite unions of étale open subsets.

Algebraically closed, real closed, p-adically closed, and bounded PAC fields are 'ez' fields.

Some Henselian and Frobenius fields are also 'ez'.

Abstract

Let $K$ be a field. The \'etale open topology on the $K$-points $V(K)$ of a $K$-variety $V$ was introduced in our previous work. The \'etale open topology is non-discrete if and only if $K$ is large. If $K$ is separably, real, $p$-adically closed then the \'etale 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 \'etale 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 \'ez fields: $K$ is \'ez if $K$ is large and every definable set is a finite union of \'etale 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 \'ez. (In particular pseudofinite fields and infinite algebraic extensions of finite fields are \'ez.) We develop the basics of a theory of definable sets in \'ez 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 \'ez.