## Notre Dame Journal of Formal Logic

### The Complexity of Primes in Computable Unique Factorization Domains

#### 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 unique factorization domain (UFD) such that the set of primes is $\Pi_{2}^{0}$-complete in every computable presentation.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 59, Number 2 (2018), 139-156.

Dates
Accepted: 20 August 2015
First available in Project Euclid: 27 February 2018

https://projecteuclid.org/euclid.ndjfl/1519722286

Digital Object Identifier
doi:10.1215/00294527-2017-0024

Mathematical Reviews number (MathSciNet)
MR3778303

Zentralblatt MATH identifier
06870284

#### Citation

Dzhafarov, Damir D.; Mileti, Joseph R. The Complexity of Primes in Computable Unique Factorization Domains. Notre Dame J. Formal Logic 59 (2018), no. 2, 139--156. doi:10.1215/00294527-2017-0024. https://projecteuclid.org/euclid.ndjfl/1519722286

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