Covering Systems in Algebraic Domains: Analogues in Number Fields and Function Fields

Classical covering systems, introduced by Erdo’s, consist of congruences ai (mod ni) whose union covers all integers. Despite extensive work on their structural and extremal properties, little is known about analogues in algebraic settings. In this paper, we develop a unified framework for covering systems over algebraic domains, focusing on the ring of integers of a number field and the polynomial ring [x]. We define algebraic covering systems in both environments and establish necessary norm and degree-based conditions for full coverage and demonstrate that restricted families of ideals or polynomial moduli cannot yield coverings unless their reciprocal norm sums exceed explicit thresholds. We further provide structural examples, and counterexamples illustrating how factorization patterns, prime splitting, and residue structure influence covering behavior. Our results show that polynomial rings admit sharper and more uniform obstruction criteria than number fields, while number-field coverings exhibit arithmetic constraints governed by prime ideal decomposition.