When is the ring of integers of a number field coverable?

TL;DR

This paper characterizes when the ring of integers of a number field can be expressed as a finite union of proper subrings, providing necessary and sufficient conditions and a formula for the minimal number of such subrings.

Contribution

It offers the first complete characterization and explicit formula for the minimal number of subrings covering the ring of integers of a number field.

Findings

Conditions based on common index divisors are necessary and sufficient.

A formula for the minimal number of subrings is provided under certain conditions.

The results connect algebraic properties of number fields with combinatorial covering properties.

Abstract

A commutative ring R is said to be coverable if it is the union of its proper subrings and said to be finitely coverable if it is the union of a finite number of them. In the latter case, we denote by {\sigma}(R) the minimal number of required subrings. In this paper, we give necessary and sufficient conditions for the ring of integers A of a given number field to be finitely coverable and a formula for {\sigma}(A) is given which holds when they are met. The conditions are expressed in terms of the existence of common index divisors and (or) common divisors of values of polynomials.