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Math Help - Show that a polynomial is irreducible but not separable over a particular field

  1. #1
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    Show that a polynomial is irreducible but not separable over a particular field

    Hi, this was a problem on an exam, and I wasn't sure how to do it.

    The problem is as follows:
    Let f(y)=y^p-x \in Z_p(x)[y], where p is prime. Show that f(y) is irreducible in Z_p(x).

    I'm lost on this one. I feel like I should assume that it's reducible, which implies that f(y) can be factored into irreducibles of lesser degree, but somehow there aren't any products of irreducibles that give us f(y). Unfortunately, I don't know how to show the "somehow" part

    Hope someone can help! Thanks!
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  2. #2
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    Re: Show that a polynomial is irreducible but not separable over a particular field

    you do know that in a field F with char(F) = p, that:

    (y - a)^p = y^p - a^p, right?

    in particular, \sqrt[p]{x} is the ONLY root of y^p - x, in any extension of \mathbb{Z}_p(x).

    so it suffices to show \sqrt[p]{x} \not \in \mathbb{Z}_p(x).

    for suppose it were. then we would have:

    x = \frac{(p(x))^p}{(q(x))^p} for some polynomials p(x),q(x) \in \mathbb{Z}_p[x].

    suppose deg(p) = k, and deg(q) = m. then deg(p(x)p) = kp, and deg(q(x)p) = mp.

    thus we have:

    1 + mp = kp

    but p divides kp, and cannot divide 1 + mp. so we have our contradiction.
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  3. #3
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    Re: Show that a polynomial is irreducible but not separable over a particular field

    Ah, yes! I do know that, I just didn't make the connection that the pth root will be the only root. Thank you!!
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