# Universally defining finitely generated subrings of global fields

## TL;DR

This paper proves that any finitely generated subring of a global field can be universally defined within its fraction field, providing a uniform proof applicable to all global fields, including characteristic two, with more efficient formulas.

## Contribution

It introduces a uniform proof for universal definability of finitely generated subrings in all global fields, including the novel characteristic two case, with simpler formulas.

## Key findings

- Universal first-order definitions exist for all finitely generated subrings of global fields.
- The proof is uniform across all global fields, including characteristic two.
- Resulting formulas require fewer quantifiers than previous methods.

## Abstract

It is shown that any finitely generated subring of a global field has a universal first-order definition in its fraction field. This covers Koenigsmann's result for the ring of integers and its subsequent extensions to rings of integers in number fields and rings of $S$-integers in global function fields of odd characteristic. In this article a proof is presented which is uniform in all global fields, including the characteristic two case, where the result is entirely novel. Furthermore, the proposed method results in universal formulae requiring significantly fewer quantifiers than the formulae that can be derived through the previous approaches.