The Complexity of Primes in Computable UFDs

TL;DR

This paper explores the computational complexity of identifying prime elements in computable unique factorization domains, demonstrating how to control prime sets' complexity while preserving factorization properties.

Contribution

It introduces methods to extend the integers computably, controlling prime set complexity in computable UFDs, and shows the existence of a UFD with a highly complex prime set.

Findings

Constructed a computable UFD with a Pi_2^0-complete prime set

Demonstrated control over prime complexity in computable extensions

Maintained unique factorization despite complex prime set

Abstract

In many simple integral domains, such as $\mathbb{Z}$ or $\mathbb{Z}[i]$, there is a straightforward procedure to determine if an element is prime by simply reducing to a direct check of finitely many potential divisors. Despite the fact that such a naive approach does not immediately translate to integral domains like $\mathbb{Z}[x]$ or the ring of integers in an algebraic number field, there still exist computational procedures that work to determine the prime elements in these cases. In contrast, we will show how to computably extend $\mathbb{Z}$ in such a way that we can control the ordinary integer primes in any $\Pi_2^0$ way, all while maintaining unique factorization. As a corollary, we establish the existence of a computable UFD such that the set of primes is $\Pi_2^0$-complete in every computable presentation.