Factorization in Domains: Applications in Error Correction

Factorization is a central computational tool in mathematics and its applications, most commonly utilized with integers and polynomials. In practice, the reliability of factorization underpins many modern technologies, with one such example being Reed-Solomon error correction. Reed-Solomon uses the factorization of polynomials to encode and recover data transmitted over noisy channels. Reed-Solomon and many algorithms like it can be understood as detecting errors by measuring how badly division fails of something that should have been a factor before the introduction of some error. This thesis examines how divisibility shapes the structure of various classifications of integral domains, focusing on the transition from Unique Factorization Domains to Euclidean domains. In Unique Factorization Domains, elements uniquely break down into prime factors, ensuring that divisibility and greatest common divisors remain well defined up to associates. Building upon this, the study explores Euclidean domains, which equip these domains with formal division algorithms. In these domains, when exact division between two elements fails, a division algorithm can be applied so that the resulting remainder is strictly bounded. In a sense, this is already enough to determine how badly division fails. However, this only identifies errors and is not sufficient for correcting them. For this reason, we build the Euclidean algorithm off of our choice of division algorithm, which enables us to pose and formally prove a generalized rational reconstruction theorem. By applying this generalized theorem directly to polynomial rings, this thesis describes how the encoding and decoding processes of the Reed-Solomon algorithm depend on the factorization properties of the underlying domains. This ultimately demonstrates how the theoretical study of abstract domains directly connects to the applied setting of modern coding theory.