Local symbols and a first-order definition of the polynomial ring over an ultra-finite field in its fraction field

TL;DR

This paper establishes a first-order definability of polynomial rings over ultraproducts of finite fields within their fraction fields, leading to undecidability results for related rational function fields.

Contribution

It introduces a universal-existential formula defining polynomial rings over ultraproducts of finite fields in their fraction fields, advancing model theory of fields.

Findings

Polynomial ring over ultraproducts definable by a first-order formula

Undecidability of the first-order theory of certain rational function fields

Use of local symbols and first-order logic in field theory

Abstract

In this paper, we prove the existence of a first-order definition of the polynomial ring over a nonprincipal ultraproduct of finite fields of unbounded cardinalities in its fraction field by a universal-existential formula in the language of rings augmented by an additional constant symbol $t$. As a consequence, we prove that the full first-order theory of the rational function field over a nonprincipal ultraproduct of finite fields of characteristic $0$ is undecidable.