Skip to main content
SpringerLink
Log in

A New Scheme for Computing with Algebraically Closed Fields

  • Conference paper
  • First Online:
Algorithmic Number Theory (ANTS 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2369))

Included in the following conference series:

Abstract

A new scheme is presented for computing with an algebraic closure of the rational field. It avoids factorization of polynomials over extension fields, but gives the illusion of a genuine field to the user. A technique of modular evaluation into a finite field ensures that a unique genuine field is simulated by the scheme and also provides fast optimizations for some critical operations. Fast modular matrix techniques are also used for several non-trivial operations. The scheme has been successfully implemented within the Magma Computer Algebra System.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Philippe Aubry, Daniel Lazard, and Marc Moreno Maza. On the theories of triangular sets. J. Symbolic Comp., 28(1–2):105–124, 1999.

    Article  MATH  Google Scholar 

  2. Thomas Becker and Volker Weispfenning. Gröbner Bases. Graduate Texts in Mathematics. Springer, New York-Berlin-Heidelberg, 1993.

    Google Scholar 

  3. Wieb Bosma and John Cannon. Handbook of Magma Functions. University of Sydney, 2001.

    Google Scholar 

  4. Wieb Bosma, John Cannon, and Catherine Playoust. The Magma Algebra System I: The User Language. J. Symbolic Comp., 24(3):235–265, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  5. David Cox, John Little, and Donal O’Shea. Ideals, Varieties and Algorithms. Undergraduate Texts in Mathematics. Springer, New York-Berlin-Heidelberg, 2nd edition, 1996.

    Google Scholar 

  6. J. Della Dora, C. Dicrescenzo, and D. Duval. About a new method for computing in algebraic number fields. In B.F. Caviness, editor, Proc. EUROCAL’ 85, volume204 of LNCS, pages 289–290, Linz, 1985. Springer.

    Google Scholar 

  7. Jean-Charles Faugère, Patrizia Gianni, Daniel Lazard, and Teo Mora. Efficient computations of zero-dimensional Gröbner bases by change of ordering. J. Symbolic Comp., 16:329–344, 1993.

    Article  MATH  Google Scholar 

  8. Donald E. Knuth. The Art of Computer Programming, volume 2. Addison Wesley, Reading, Massachusetts, 3rd edition, 1997.

    Google Scholar 

  9. Daniel Lazard. Solving zero-dimensional algebraic systems. J. Symbolic Comp., 13(2):117–131, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  10. Magma Computational Algebra System: Homepage. University of Sydney. URL:http://magma.maths.usyd.edu.au.

  11. Barry M. Trager. Algebraic factoring and rational function integration. In R.D. Jenks, editor, Proc. SYMSAC’ 76, pages 196–208. ACM press, 1976.

    Google Scholar 

  12. Mark van Hoeij. Factoring polynomials and the knapsack problem. J. Number Th., to appear 2002. URL:http://www.math.fsu.edu/~hoeij/paper/knapsack.ps.

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, University of Sydney, NSW, 2006, Australia

    Allan Steel

Authors
  1. Allan Steel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Mathematics and Statistics, F07, University of Sydney, Sydney, NSW, 2006, Australia

    Claus Fieker  & David R. Kohel  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Steel, A. (2002). A New Scheme for Computing with Algebraically Closed Fields. In: Fieker, C., Kohel, D.R. (eds) Algorithmic Number Theory. ANTS 2002. Lecture Notes in Computer Science, vol 2369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45455-1_38

Download citation

  • DOI: https://doi.org/10.1007/3-540-45455-1_38

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43863-2

  • Online ISBN: 978-3-540-45455-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation