Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 59, Number 2 (2018), 139-156.
The Complexity of Primes in Computable Unique Factorization Domains
In many simple integral domains, such as or , 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 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 in such a way that we can control the ordinary integer primes in any 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 -complete in every computable presentation.
Notre Dame J. Formal Logic, Volume 59, Number 2 (2018), 139-156.
Received: 19 November 2014
Accepted: 20 August 2015
First available in Project Euclid: 27 February 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 03C57: Effective and recursion-theoretic model theory [See also 03D45] 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55]
Secondary: 13F15: Rings defined by factorization properties (e.g., atomic, factorial, half- factorial) [See also 13A05, 14M05] 13L05: Applications of logic to commutative algebra [See also 03Cxx, 03Hxx]
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