Notre Dame Journal of Formal Logic

The Complexity of Primes in Computable Unique Factorization Domains

Damir D. Dzhafarov and Joseph R. Mileti

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1519722286

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

Mathematical Reviews number (MathSciNet)
MR3778303

Zentralblatt MATH identifier
06870284

Subjects
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]

Keywords
computable unique factorization domains computability theory primes

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


Export citation

References

  • [1] Baur, W., “Rekursive Algebren mit Kettenbedingungen,” Mathematical Logic Quarterly, vol. 20 (1974), pp. 37–46.
  • [2] Cohen, H., A Course in Computational Algebraic Number Theory, vol. 138 of Graduate Texts in Mathematics, Springer, Berlin, 1993.
  • [3] Cohen, H., Advanced Topics in Computational Number Theory, vol. 193 of Graduate Texts in Mathematics, Springer, New York, 2000.
  • [4] Conidis, C. J., “On the complexity of radicals in noncommutative rings,” Journal of Algebra, vol. 322 (2009), pp. 3670–80.
  • [5] Conrad, K., “Factoring in quadratic fields,” preprint, http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/quadraticgrad.pdf (accessed 4 October 2017).
  • [6] Crandall, R., and C. Pomerance, Prime Numbers: A Computational Perspective, 2nd edition, Springer, New York, 2005.
  • [7] Downey, R. G., and A. M. Kach, “Euclidean functions of computable Euclidean domains,” Notre Dame Journal of Formal Logic, vol. 52 (2011), pp. 163–72.
  • [8] Downey, R. G., S. Lempp, and J. R. Mileti, “Ideals in computable rings,” Journal of Algebra, vol. 314 (2007), pp. 872–87.
  • [9] Eisenbud, D., Commutative Algebra: With a View Toward Algebraic Geometry, vol. 150 of Graduate Texts in Mathematics, Springer, New York, 1995.
  • [10] Friedman, H. M., S. G. Simpson, and R. L. Smith, “Countable algebra and set existence axioms,” Annals of Pure and Applied Logic, vol. 25 (1983), pp. 141–81. Addendum, Annals of Pure and Applied Logic, vol. 28 (1985), pp. 319–20.
  • [11] Fröhlich, A., and J. C. Shepherdson, “Effective procedures in field theory,” Philosophical Transactions of the Royal Society of London, Series A, vol. 248 (1956), pp. 407–32.
  • [12] Klüners, J., “Algorithms for function fields,” Experimental Mathematics, vol. 11 (2002), pp. 171–81.
  • [13] Matsumura, H., Commutative Ring Theory, vol. 8 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1986.
  • [14] Metakides, G., and A. Nerode, “Effective content of field theory,” Annals of Mathematical Logic, vol. 17 (1979), pp. 289–320.
  • [15] Miller, R., “Computable fields and Galois theory,” Notices of the American Mathematical Society, vol. 55 (2008), pp. 798–807.
  • [16] Müller-Quade, J., and R. Steinwandt, “Basic algorithms for rational function fields,” Journal of Symbolic Computation, vol. 27 (1999), pp. 143–70.
  • [17] Rabin, M. O., “Computable algebra, general theory and theory of computable fields,” Transactions of the American Mathematical Society, vol. 95 (1960), pp. 341–60.
  • [18] Soare, R. I., Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets, Perspectives in Mathematical Logic, Springer, Berlin, 1987.
  • [19] Stoltenberg-Hansen, V., and J. V. Tucker, “Computable rings and fields,” pp. 363–447 in Handbook of Computability Theory, vol. 140 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1999.