axịom™ pp 227-307 | Cite as

Advanced Problem Solving

  • Richard D. Jenks
  • Robert S. Sutor


In this chapter we describe techniques useful in solving advanced problems with AXIOM.


Power Series Finite Field Galois Group Algebraic Number Primitive Element 
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  1. 1.
    For more information about the algebraic structure and properties of finite fields, see, for example, S. Lang, Algebra, Second Edition, New York: Addison-Wesley Publishing Company, Inc., 1984, ISBN 0 201 05487 6MATHGoogle Scholar
  2. R. Lidl, H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and Its Applications, Vol. 20, Cambridge: Cambridge Univ. Press, 1983, ISBN 0 521 30240 4.MATHGoogle Scholar
  3. 2.
    For For more information on the implementation aspects of finite fields, see J. Grabmeier, A. Scheerhorn, Finite Fields in Axiom, Technical Report, IBM Heidelberg Scientific Center, 1992.Google Scholar
  4. 4.
    Cf. Lidl, R. & Niederreiter, H., Finite Fields, Encycl. of Math. 20, (Addison-Wesley, 1983), p.90, Th. 3.18.MATHGoogle Scholar
  5. 5.
    The existence of such polynomials is proved in Lenstra, H. W. & Schoof, R. J., Primitive Normal Bases for Finite Fields, Math. Comp. 48, 1987, pp. 217–231.CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    See McKay, Soicher, Computing Galois Groups over the Rationals, Journal of Number Theory 20, 273–281 (1983). We do not assume the results of this paper, however, and we continue with the computation.MathSciNetGoogle Scholar
  7. 9.
    The interested reader can learn more about these aspects of the AXIOM library from the paper “Computations in Algebras of Finite Rank,” by Johannes Grabmeier and Robert Wisbauer, Technical Report, IBM Heidelberg Scientific Center, 1992.Google Scholar
  8. 10.
    Worz-Busekros, A., Algebras in Genetics, Springer Lectures Notes in Biomathematics 36, Berlin e.a. (1980). In particular, see example 1.3.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Richard D. Jenks
    • 1
  • Robert S. Sutor
    • 1
  1. 1.IBM Thomas J. Watson Research CenterYorktown HeightsUSA

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