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Math Help - Using eisenstein

  1. #1
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    Using eisenstein

    I know how to prove this for a general case using Eisenstein criterion but not for all of p and q

    If p and q are prime numbers and p does not = q, then x^5-p^5*q contained in Z(x) is irreducible in Q(x).
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  2. #2
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    Quote Originally Posted by chadlyter View Post
    I know how to prove this for a general case using Eisenstein criterion but not for all of p and q

    If p and q are prime numbers and p does not = q, then x^5-p^5*q contained in Z(x) is irreducible in Q(x).
    It is necessary and sufficient to show that x^5-p^5*q is irreducible in Z[x] for that will imply that it is irreducible in Q[x].

    Use Eisenstein Criterion with q, then q|p^5*q obviously.
    But q^2 does not divide p^5*q since p!=q.
    Q.E.D.
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