Soto Rosa, Geraldo E.

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    Computable finite factorization domains
    (2023-05-10) Soto Rosa, Geraldo E.; Ocasio González, Victor A.; College of Arts and Sciences - Sciences; Ortiz Albino, Reyes M.; Cáceres Duque, Luis F.; Department of Mathematics; Alers Valentín, Hilton
    In the ring of integers Z we can obtain the set of all the divisors of a given number using the Division Algorithm. Moreover, since there are finitely many divisors for each element, the Kronecker Method allows us to construct an algorithm to find the set of all divisors for polynomials in Z[x]. In both cases, the existence of these algorithms implies that the set of irreducible elements is computable relative to the copy of the structure. Other integral domains, like the quadratic extensions of Z, also have this property. In general, these domains are examples of Strongly Computable Strong Finite Factorization Domains (SCSFFD). In 2017, these structures were used to prove the existence of an integral domain where the set of irreducible elements is computable while the set of prime elements is not. Although differences are known between irreducible and prime elements in the algebraic context, this result shows how different they can be in computability context. Also, the authors provided conditions for which a Unique Factorization Domain (UFD) is a SCSFFD. However, being an UFD is not sufficient because there exists integral domains like Z[\sqrt{-5}] that are SCSFFD but not UFD. Our work completely classifies SCSFFD's in general by showing the existence of a computable norm that possess among other properties, being able to solve norm-form equations computably. This classification provides the intuition to extend further the notion of strongly computability to Computable Finite Factorization Domains in general.