Abstract
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.
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References
Factoring polynomials with rational coefficients, Math. Ann., 261, 1982, p. 515–534.
Factoring Polynomials with Rational Coefficients, Math. Ann., 261, 1982, p. 515–534.
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).
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.
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.
A more efficient Algorithm for Lattice Basis Reduction, J. Algorithms, 9, 1988, p. 47–62.
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.
This result is due to W. Specht. — Zur Theorie der algebraischen Gleichungen, Jahresber. Deutsch. Math., 5, 1938, p. 142–145.
For a more detailed proof see, for example, Maxim Bôcher, Introduction to higher algebra, Maemillan, New York, 1907.
B.R. McDonald. — Linear Algebra over commutative rings; Marcel Dekker, New York, 1984.
G.B. Price. — Some identities in the theory of determinants; Amer. Math. Monthly, 54, 1947, p. 75–90.
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© 1992 Springer-Verlag New York, Inc.
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Mignotte, M. (1992). Polynomials with Integer Coefficients. In: Mathematics for Computer Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9171-5_7
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