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Factorization of Polynomials over a field

  1. Jan 27, 2009 #1
    I don't understand how to factor a polynomial over Z3 [x], Z7 [x], and Z11 [x]

    I need to factor the polynomail x3 - 23x2 - 97x + 291

    Last edited: Jan 28, 2009
  2. jcsd
  3. Jan 28, 2009 #2
    Note that for a polynomial is of degree 2 and 3, reducibility is equivalent to the existence of roots.

    mod 3:

    X^3-23X^2-97X+291=X^3+X^2+2X=X(X^2+X+2). A calculation shows X^2+X+2 doesn't have a root in Z/3Z. Done.

    mod 7:
    X^3+5X^2+2X+4. A calculation shows it has no root in Z/7Z. The polynomial is irreducible.

    mod 11:
    291=5=1*5=10*6. So if the polynomial has a root, it should be 1, 5, 6, or 10. A calculation shows X^3-X^2+2X+5 has no root. The given poly is irreducible.
  4. Jan 28, 2009 #3
    I think I might have phrased this question wrong, but I figured it out.

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