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arXiv · 2306.12566

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

Mohamed Ayad·Omar Kihel
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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.

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Mohamed Ayad, Omar Kihel. 2023-06-08. When is the ring of integers of a number field coverable?. https://arxiv.org/abs/2306.12566

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