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Group theory problem

  1. Jan 25, 2008 #1
    [SOLVED] group theory problem

    1. The problem statement, all variables and given/known data
    Find all solutions of the equation x^3-2x^2-3x=0 in Z_12.


    2. Relevant equations



    3. The attempt at a solution
    We first factor the polynomial into x(x-3)(x+1)=0. Recall that Z_12 is not an integral domain since 12 is not prime (e.g. 3*4=0). Therefore setting each factor equal to 0 WILL NOT GIVE ALL OF THE SOLUTIONS.

    Obviously, the solutions to x=0, (x-3)=0, (x+1)=0, x(x-3)=0, x(x+1) = 0, (x-3)(x+1)=0 will also be solutions to our equation. I can find all of those. The problem is that I do not know how to find the remaining ones.
     
  2. jcsd
  3. Jan 25, 2008 #2

    Hurkyl

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    You could narrow things down by factoring 12 into prime powers, and using the chinese remainder theorem.

    You can narrow things down even further in Z/4Z by first considering it in Z/2Z.


    Or... you could apply the fact that each solutions will make at least one of the factors a zero divisor..


    But honestly, 12 is so small that I'd expect simply trying all 12 possibilities is the most efficient way to find the roots.
     
    Last edited: Jan 25, 2008
  4. Jan 25, 2008 #3
    What are the twelve possibilities?
     
  5. Jan 25, 2008 #4

    Dick

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    x=0,1,2...11. What else??
     
  6. Jan 25, 2008 #5

    HallsofIvy

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    From what you said before, it would appear that you know what Z12 is! The" 12 possibilities" Hurkyl mentioned are the 12 elements of that ring.
     
  7. Jan 25, 2008 #6
    Grrrr. Someday I will stop making mistakes like this.
     
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