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Math Help - Solution to equation in field

  1. #1
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    Solution to equation in field

    Find all solutions to the equation x^{16} = 1 in the field Z_{17}.

    A solution or a starting point would be greatly appreciated.
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  2. #2
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    Quote Originally Posted by featherbox View Post
    Find all solutions to the equation x^{16} = 1 in the field Z_{17}.

    A solution or a starting point would be greatly appreciated.
    Since \mathbb{Z}_{17}-{0} is a finite group of order 16 , so for every nonzero x\in \mathbb{Z}_{17} we have x^{16} = 1 , so the solution is \mathbb{Z}_{17}-{0} .
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  3. #3
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    I got that because 16=-1 in the field, we have the equation now being x^{-1}=1, and seeing as all elements in the field have multiplicative inverses all elements within Z_{17} are solutions.
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    Quote Originally Posted by featherbox View Post
    I got that because 16=-1 in the field, we have the equation now being x^{-1}=1, and seeing as all elements in the field have multiplicative inverses all elements within Z_{17} are solutions.
    you can't say x^{16}=x^{-1} , as I said x^{16}=1 and from x^{-1}=1 you can just conclude that x=1 and nothing else !
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  5. #5
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    Quote Originally Posted by xixi View Post
    you can't say x^{16}=x^{-1} , as I said x^{16}=1 and from x^{-1}=1 you can just conclude that x=1 and nothing else !
    Thats not what I said at all.
    The solutions would be 1,2,...16. NOT just 1.
    This is because x^{-1}=1 for x=1,2...,16

    Why can't x^{16}=x^{-1}?
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  6. #6
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    Quote Originally Posted by featherbox View Post
    Thats not what I said at all.
    The solutions would be 1,2,...16. NOT just 1.
    This is because x^{-1}=1 for x=1,2...,16

    Why can't x^{16}=x^{-1}?
    from x^{-1}=1 we have x=1 because the only element whose inverse is 1 is itself 1 . Moreover since the order of each element divides the order of the group and the order of this group is 16 then x^{16}=1 and x^{16}=x^{17}x^{-1} but x^{17}=x and so x^{16}=x.x^{-1}=1 and as you see it won't be x^{-1}.
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