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Math Help - Trying to prove or disprove statement about irreducibility

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    Trying to prove or disprove statement about irreducibility

    This is the statement I am trying to prove or disprove:

    Let p(x) = \frac{x^m - 1}{x - 1} where m = p^n for some prime number p and some positive integer n. Then p(x) has exactly n irreducible factors over the integers.

    I can see how such polynomials could be factored into n factors but I am having difficulty proving or disproving that the factors are irreducible.

    For instance, if p is 2 and n is 4 we have:

    p(x) = \frac{x^{16} - 1}{x - 1} = (x^8 + 1)(x^4 + 1)(x^2 + 1)(x + 1)

    Clearly these factors are irreducible.

    If p is 3 and n is 3 we have:

    p(x) = \frac{x^{27} - 1}{x - 1} = (x^{18} + x^9 + 1)(x^6 + x^3 + 1)(x^2 + x + 1)

    If p is 5 and n is 3 we have:

    p(x) = \frac{x^{125} - 1}{x - 1} = (x^{100} + x^{75} + x^{50} + x^{25} + 1)(x^{20} + x^{15} + x^{10} + x^5 + 1)(x^4 + x^3 + x^2 + x + 1)

    So I see the pattern that produces these factors, but I don't know how to show that all of them are irreducible if they are.
    Last edited by icemanfan; January 16th 2012 at 03:49 PM.
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