Notre Dame Journal of Formal Logic

The Complexity of Primes in Computable Unique Factorization Domains

Damir D. Dzhafarov and Joseph R. Mileti

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In many simple integral domains, such as Z or 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 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 Z in such a way that we can control the ordinary integer primes in any Π20 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 Π20-complete in every computable presentation.

Article information

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

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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]

computable unique factorization domains computability theory primes


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.

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