Denseness results in the theory of algebraic fields

TL;DR

This paper investigates the conditions under which algebraic fields are dense in their real and p-adic closures, showing that this property is elementary and holds for all models of the theory of algebraic fields.

Contribution

It proves that density in real and p-adic closures is an elementary property in the language of rings for algebraic fields.

Findings

All models of the theory of algebraic fields are dense in their real and p-adic closures.

The property of denseness is elementary in the language of rings.

The paper establishes a connection between algebraic field properties and model-theoretic elementary classes.

Abstract

We study when the property that a field is dense in its real and p-adic closures is elementary in the language of rings and deduce that all models of the theory of algebraic fields have this property.