Skip to main content
SpringerLink
Log in

An algorithm for computing the factor ring of an ideal in Dedekind domain with finite rank

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We give an algorithm for computing the factor ring of a given ideal in a Dedekind domain with finite rank, which runs in deterministic and polynomial-time. We provide two applications of the algorithm: judging whether a given ideal is prime or prime power. The main algorithm is based on basis representation of finite rings which is computed via Hermite and Smith normal forms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agrawal M, Kayal N, Saxena N. PRIMES is in P. Ann of Math (2), 2004, 160: 781–793

    Article  MathSciNet  MATH  Google Scholar 

  2. Arvind V, Das B, Mukhopadhyay P. The complexity of black-box ring problems. In: Computing and Combinatorics. Berlin-Heidelberg: Springer, 2006, 126–135

    Chapter  Google Scholar 

  3. Atiyah M F, Macdonald I G. Introduction to Commutative Algebra. Reading: Addison-Wesley, 1969

    MATH  Google Scholar 

  4. Cohen H. A Course in Computational Algebraic Number Theory, Third Corrected Printing. New York: Springer, 1996

    Google Scholar 

  5. Cohen H. Advanced Topics in Computational Number Theory. New York: Springer, 2000

    Book  MATH  Google Scholar 

  6. Hafner J, McCurley K. Asymptotically fast triangularization of matrices over rings. SIAM J Comput, 1991, 20: 1068–1083

    Article  MathSciNet  MATH  Google Scholar 

  7. Hermite C. Sur l′introduction des variables continues dans la théorie des nombres. J Reine Angew Math, 1851, 41: 191–216

    Article  MathSciNet  Google Scholar 

  8. Iliopoulos C. Worst-case complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the Hermite and Smith normal forms of an integer matrix. SIAM J Comput, 1989, 18: 658–669

    Article  MathSciNet  MATH  Google Scholar 

  9. Janusz G. Algebraic Number Fields, 2nd ed. Graduate Studies in Mathematics, vol. 7. Providence: Amer Math Soc, 1996

    MATH  Google Scholar 

  10. Kayal N, Saxena N. Complexity of ring morphism problems. Comput Complexity, 2006, 15: 342–390

    Article  MathSciNet  MATH  Google Scholar 

  11. Lenstra H W. Algorithms in algebraic number theory. Bull Amer Math Soc (NS), 1992, 26: 211–244

    Article  MathSciNet  MATH  Google Scholar 

  12. Schönhage A, Strassen V. Schnelle multiplikation grosser zahlen. Computing, 1971, 7: 281–292

    Article  MathSciNet  MATH  Google Scholar 

  13. Shoup V. Fast construction of irreducible polynomials over finite fields. J Symbolic Comput, 1994, 17: 371–391

    Article  MathSciNet  MATH  Google Scholar 

  14. Smith H J S. On systems of linear indeterminate equations and congruences. Philos Trans R Soc Lond Ser A Math Phys Eng Sci, 1861, 151: 293–326

    Article  Google Scholar 

  15. Staromiejski M. Polynomial-time locality tests for finite rings. J Algebra, 2013, 379: 441–452

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11601202, 11471314 and 11401312), the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant No. 14KJB110012), the High-Level Talent Scientific Research Foundation of Jinling Institute of Technology (Grant No. jit-b-201527), and the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences. The authors thank Dr. Yunling Kang for his helpful discussion.

Author information

Authors and Affiliations

  1. Department of Cyber Space Security, School of Software Engineering, Jinling Institute of Technology, Nanjing, 211169, China

    Dandan Huang

  2. Key Laboratory of Mathematics Mechanization, National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Yingpu Deng

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Yingpu Deng

Authors
  1. Dandan Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yingpu Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dandan Huang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, D., Deng, Y. An algorithm for computing the factor ring of an ideal in Dedekind domain with finite rank. Sci. China Math. 61, 783–796 (2018). https://doi.org/10.1007/s11425-016-9060-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-016-9060-2

Keywords

MSC(2010)

Search

Navigation