Polynomials with Integer Coefficients

  • Maurice Mignotte


This chapter is the culmination of this book. Here we describe algorithms of factorization of polynomials with integer coefficients. These algorithms use many of the results demonstrated in the previous chapters.


Prime Number Previous Chapter Chinese Remainder Theorem Irreducible Factor Integer Coefficient 
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  1. *.
    Factoring polynomials with rational coefficients, Math. Ann., 261, 1982, p. 515–534.Google Scholar
  2. †.
    Factoring Polynomials with Rational Coefficients, Math. Ann., 261, 1982, p. 515–534.Google Scholar
  3. ∥.
    See, for example, J.W.S. Casseis.— An introduction to the geometry of numbers, Springer, Heidelberg, 1971 (Lem. 4, Ch. 1, and Th. 1, Ch. 2).Google Scholar
  4. †.
    On the Complexity of Finding Short Vectors in Integer Lattices, EUROCAL’ 83, London, ed. by van Hulzen, Lecture Notes in Computer Science, N. 162, Springer, Heidelberg, 1983.Google Scholar
  5. ‡.
    Factorization of univariate integer polynomials by diophantine approximation and an improved basis reduction algorithm, Proc. of 11th ICALP, Antwerpen, 1984, Lecture Notes in Computer Science, N. 172, Springer, Heidelberg, 1984.Google Scholar
  6. ¶.
    A more efficient Algorithm for Lattice Basis Reduction, J. Algorithms, 9, 1988, p. 47–62.Google Scholar
  7. †.
    This examples is due to H.P.F. Swinnerton-Dyer. See the article of E. Kaltofen, Factorization of Polynomials in Computer Algebra, ed. B. Buchberger, G. E. Collins, R. Loos, Springer, 1982, p. 95–113.Google Scholar
  8. †.
    This result is due to W. Specht. — Zur Theorie der algebraischen Gleichungen, Jahresber. Deutsch. Math., 5, 1938, p. 142–145.Google Scholar
  9. ‡.
    For a more detailed proof see, for example, Maxim Bôcher, Introduction to higher algebra, Maemillan, New York, 1907.MATHGoogle Scholar
  10. ∥.
    B.R. McDonald. — Linear Algebra over commutative rings; Marcel Dekker, New York, 1984.MATHGoogle Scholar
  11. §.
    G.B. Price. — Some identities in the theory of determinants; Amer. Math. Monthly, 54, 1947, p. 75–90.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • Maurice Mignotte
    • 1
  1. 1.Départment de MathématiqueUniversité Louis PasteurStrasbourgFrance

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