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Number theory. If k divides p - 1, show that the congruence has exactly k distinct so

  1. Apr 4, 2012 #1
    1. The problem statement, all variables and given/known data

    http://i43.tinypic.com/fymy3l.jpg

    question 22.4 (a)

    2. Relevant equations



    3. The attempt at a solution

    xk=(xp-1)m = (xp-1-1)(1 + xp-1 +x2(p-1) +...+x(m-1)(p-1))

    I know that xp-1-1 = 0 mod p has p-1 solutions but I can't make anything from the geometric sum. Can someone push me in the right direction.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 4, 2012 #2

    SammyS

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    Re: Number theory. If k divides p - 1, show that the congruence has exactly k distinc

    (xp-1-1)(1 + xp-1 +x2(p-1) +...+x(m-1)(p-1)) ≠ (xp-1)m

    (xp-1-1)(1 + xp-1 +x2(p-1) +...+x(m-1)(p-1)) = (xp-1)m - 1
     
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