A universal first order formula defining the ring of integers in a number field

TL;DR

This paper proves that the set of integers in a number field can be characterized by a universal first-order formula, extending previous results from the rational numbers to general number fields using class field theory.

Contribution

It introduces a universal first-order formula defining the ring of integers in any number field, generalizing Koenigsmann's result from Q to all number fields.

Findings

The complement of the ring of integers in a number field is Diophantine.

A universal first-order formula characterizes the ring of integers in number fields.

The approach uses global class field theory.

Abstract

We show that the complement of the ring of integers in a number field K is Diophantine. This means the set of ring of integers in K can be written as {t in K | for all x_1, ..., x_N in K, f(t,x_1, ..., x_N) is not 0}. We will use global class field theory and generalize the ideas originating from Koenigsmann's recent result giving a universal first order formula for Z in Q.