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Math Help - Factorization (using eisenstein)

  1. #1
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    Factorization (using eisenstein)

    Hi,

    Can someone please help me with the following question and explain to me what they are doing at each step:

    In each case below, decide whether or not the given polynomial is irreducible over Q, justifying your answer in each case. If the polynomial is not irreducible , give its complete factorization into Q-irreducible factors.

    i) (x^12) - 18(x^6) - 175

    ii) (x^6) + (x^3) + 1

    iii) (x^10) + 1


    Thanks.
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  2. #2
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    A polynomial is reducibble in Z[x] if and only if it is reducible in Q[x].

    Quote Originally Posted by jedoob View Post

    i) (x^12) - 18(x^6) - 175
    Begin this problem by writing,
    y^2-18y-175=0 where y=x^6.
    Find zeros and hence find the factors,
    (y-25)(y+7)=0
    Thus,
    (x^3-25)(x^3+7)=0
    Those two polyonomial are irreducible in Z[x].
    Because,
    x^3-25 has no zero (and has degree at most three).
    x^3+7 has no zero (and has degree at most three).
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