Irreducibles and primes in computable integral domains

TL;DR

This paper explores the computational complexity of prime and irreducible elements in computable integral domains, showing they can differ significantly and generalizing classical factorization methods.

Contribution

It constructs computable integral domains with non-computably related prime and irreducible sets and generalizes Kronecker's method for polynomial factorization.

Findings

Existence of computable UFDs with non-computable prime/irreducible sets

Construction of domains where irreducibles are computable but primes are not, and vice versa

Generalization of Kronecker's method for $\\mathbb{Z}[x]$

Abstract

A computable ring is a ring equipped with mechanical procedure to add and multiply elements. In most natural computable integral domains, there is a computational procedure to determine if a given element is prime/irreducible. However, there do exist computable UFDs (in fact, polynomial rings over computable fields) where the set of prime/irreducible elements is not computable. Outside of the class of UFDs, the notions of irreducible and prime may not coincide. We demonstrate how different these concepts can be by constructing computable integral domains where the set of irreducible elements is computable while the set of prime elements is not, and vice versa. Along the way, we will generalize Kronecker's method for computing irreducibles and factorizations in $\mathbb{Z}[x]$.