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Math Help - modular arithmetic

  1. #1
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    modular arithmetic

    What are the solutions to x^2+1=0 in Z_3, Z_2 and C. Is the answer as simple as: no solutions in Z_3 and Z_2 and x=i,-i in C?
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  2. #2
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    Re: modular arithmetic

    short answer: no.

    slightly longer answer: x^2+1 has a solution in Z2. find it.
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  3. #3
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    Re: modular arithmetic

    but -1 isn't a member of Z2.
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  4. #4
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    Re: modular arithmetic

    are you sure about that? what is 1 + 1 in Z2?
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  5. #5
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    Re: modular arithmetic

    0
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  6. #6
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    Re: modular arithmetic

    Hello, Duke!

    x^2+1\:\equiv\:0\text{ (mod 3)}

    We have: . x^2 + 1 \:\equiv\:0\text{ (mod 3)}

    . . . . . . . . . . . x^2 \:\equiv\:\text{-}1\text{ (mod 3)}

    . . . . . . . . . . . x^2 \:\equiv\:2\text{ (mod 3)}\;\;\bf{[1]}


    We find that: . \begin{Bmatrix} 0^2 &\equiv& 0 \text{ (mod 3)} \\ 1^2 &\equiv& 1\text{ (mod 3)} \\ 2^2 &\equiv& 1\text{ (mod 3)} \end{Bmatrix}

    Therefore, \bf{[1]} has no solutions.




    x^2+1\:\equiv\:0\text{ (mod 2)}

    We have: . x^2 + 1 \:\equiv\:0\text{ (mod 2)}

    . . . . . . . . . . . x^2 \:\equiv\:\text{-}1\text{ (mod 2)}

    . . . . . . . . . . . x^2 \:\equiv\:1\text{ (mod 2)}\;\;\bf{[2]}


    We find that: . \begin{Bmatrix} 0^2 &=& 0 \text{ (mod 2)} \\ 1^2 &\equiv& 1\text{ (mod 2)} \end{Bmatrix}

    Therefore, \bf{[2]} has one solution: x \:\equiv\:1\text{ (mod 2)}




    x^2 + 1 \:\equiv\: 0 \text{ (mod }C)

    We have: . x^2 + 1 \:\equiv\:0\text{ (mod }C}

    . . . . . . . . . . . x^2 \:\equiv\:\text{-}1\text{ (mod }C)

    . . . . . . . . . . . x^2 \:\equiv\:C-1\text{ (mod }C)\;\;{\bf{[3]}


    \bf{[3]} has solutions if C \,=\,k^2+1 for a positive integer k.

    Are there any other solutions?
    . . I don't know.

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  7. #7
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    Re: modular arithmetic

    Thank you for revealing my total ignorance of modular arithmetic.
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