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Quotient Ring Proof

  1. Nov 18, 2009 #1
    1. The problem statement, all variables and given/known data

    For a prime p and a polynomial g(x) that is irreducible in [tex]Z_{p}[X][/tex], prove that for any f(x) in [tex]Z_{p}[X][/tex] and integer k > 1, [tex][f(x)]^{k} = [f(x)][/tex] in [tex]Z_{p}[X]/(g(x))[/tex].


    3. The attempt at a solution

    I realize this is an extension of Fermat's Little Theorem, however I cannot figure out how to proceed. I attempted to adapt a proof of FLT for the integers modulo p, but couldn't get anywhere. Any nudges in the right direction would be greatly appreciated!
     
  2. jcsd
  3. Nov 18, 2009 #2

    Hurkyl

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    Did you mean "there exists an integer k > 1" instead of "for every integer k > 1"?


    What is the proof of FLT you're trying to adapt? What part of it doesn't work?
     
  4. Nov 18, 2009 #3
    Yes, Hurkyl, I meant there exists an integer k > 1. Sorry about that.
     
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