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Math Help - irreducible

  1. #1
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    irreducible

    Prove that x^{p^n}-x+1 is irreducible over \mathbb{F}_p only when n=1 or n=p=2.
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  2. #2
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    Quote Originally Posted by Stiger View Post
    Prove that x^{p^n}-x+1 is irreducible over \mathbb{F}_p only when n=1 or n=p=2.
    if n=p^km, where m > 1, \ k \geq 0, \ \gcd(p,m)=1, then it's fairly easy to see that x^{p^{p^k}} - x + m^{-1} will divide x^{p^n}-x+1. so we may assume that n=p^k, \ k \geq 0.
    Last edited by NonCommAlg; December 10th 2008 at 12:08 AM.
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    if n=p^km, where m > 1, \ k \geq 0, \ \gcd(p,m)=1, then it's fairly easy to see that x^{p^{p^k}} - x + m^{-1} will divide x^{p^n}-x+1. so we may assume that n=p^k, \ k \geq 0.
    Why does x^{p^{p^k}} - x + m^{-1} divide x^{p^n}-x+1.?
    Please...give me some more detail explanation.
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