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Math Help - Separable polynomial over field of characteristic 0

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    Ant
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    Separable polynomial over field of characteristic 0

    Let K be a field of characteristic 0. Let f \in K[X] be an irreducible polynomial. I want to prove that f has no repeated roots in its splitting field. i.e. I want to show that it is separable.

    It we let  f' denote the formal derivative of f.

    So assume (for contradiction) that f is not separable. Then it has a double root in it's splitting field, \Sigma_f, it we call this root \alpha \in \Sigma_f . Then  f, f' have a common divisor of degree at least 1 in \Sigma_f[X], let's call it g(X).

    I get that if this common divisor, g(X) is actually in K[X] we get a contraction (because f(X) is irreducible), but what if g(X) \in \Sigma_f [X] \setminus K[X]?

    Could anyone help?
    Last edited by Ant; May 23rd 2013 at 02:52 AM.
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