Definability and Decidability in Infinite Algebraic Extensions

TL;DR

This paper demonstrates the existence of fields with specific definability properties and explores the undecidability of certain field theories, extending understanding of logical definability and decidability in algebraic structures.

Contribution

It generalizes Ziegler's construction to produce fields with definable subsets and analyzes the undecidability of non-algebraically closed field theories.

Findings

Existence of fields with definable subsets for any countable collection.

Many infinitely axiomatizable field theories are finitely hereditarily undecidable.

Construction method applicable to fields with positive transcendence degree.

Abstract

We use a generalization of a construction by Ziegler to show that for any field $F$ and any countable collection of countable subsets $A_i \subseteq F, i \in \calI \subset \Z_{>0}$ there exist infinitely many fields $K$ of arbitrary positive transcendence degree over $F$ and of infinite algebraic degree such that each $A_i$ is first-order definable over $K$. We also use the construction to show that many infinitely axiomatizable theories of fields which are not compatible with the theory of algebraically closed fields are finitely hereditarily undecidable.