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2018 The Complexity of Primes in Computable Unique Factorization Domains
Damir D. Dzhafarov, Joseph R. Mileti
Notre Dame J. Formal Logic 59(2): 139-156 (2018). DOI: 10.1215/00294527-2017-0024


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.


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Damir D. Dzhafarov. Joseph R. Mileti. "The Complexity of Primes in Computable Unique Factorization Domains." Notre Dame J. Formal Logic 59 (2) 139 - 156, 2018.


Received: 19 November 2014; Accepted: 20 August 2015; Published: 2018
First available in Project Euclid: 27 February 2018

zbMATH: 06870284
MathSciNet: MR3778303
Digital Object Identifier: 10.1215/00294527-2017-0024

Primary: 03C57 , 03D45
Secondary: 13F15 , 13L05

Keywords: computability theory , computable unique factorization domains , primes

Rights: Copyright © 2018 University of Notre Dame


Vol.59 • No. 2 • 2018
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