Exact Covering Systems in Number Fields

TL;DR

This paper proves Kim's conjecture that in any algebraic number field, exact covering systems must have repeated moduli, extending known results from integers and quadratic fields to all algebraic number fields.

Contribution

The paper establishes that exact covering systems in any algebraic number field necessarily have repeated moduli, confirming Kim's conjecture.

Findings

Exact covering systems in algebraic number fields must have repeated moduli

Kim's conjecture is proven for all algebraic number fields

Extends previous results from integers and quadratic fields

Abstract

It is well known that in an exact covering system in $\mathbb{Z}$, the biggest modulus must be repeated. Very recently, Kim gave an analogous result for certain quadratic fields, and Kim also conjectured that it must hold in any algebraic number field. In this paper, we prove Kim's conjecture. In other words, we prove that exact covering systems in any algebraic number field must have repeated moduli.