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Math Help - a question on field extension

  1. #1
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    a question on field extension

    Let E/F an algebraic field extension and let p=CharF>0 .
    Let a be an elemnt in E.
    prove that there exists i>=0 such as a^p^i is separable.
    i thought to use to fact that if F is a field with Char p >0, and
    f(x)=g(x^p) for some g over F, then f(x) (which its root is a) isn't separable..
    but i'm not quite sure how to use it...

    Thanks in advance
    Last edited by hod87; May 18th 2011 at 11:18 AM.
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  2. #2
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    You're on the right track. If f(x) is not separable, then write f(x) = g(x^p). If g(x) is separable then we are done. If not, we can repeat the process, writing g(x) as some h(x^{p_2}). Eventually you have to get a separable polynomial, and the result follows immediately.
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