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Math Help - Congruence in Z

  1. #1
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    Congruence in Z

    Let p be prime in Z with  p \equiv 1 \ (mod \ 4). Show that there is an integer  n \in Z with  n^2 \equiv -1 \ (mod \ p)

    Proof. Now I have p - 1 = 4k for some integers k. Then p = 4k - 1, pk = 4k^2 - k, so I have a square of something but with an extra k behind, how can I continue?

    Thank you.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Let p be prime in Z with  p \equiv 1 \ (mod \ 4). Show that there is an integer  n \in Z with  n^2 \equiv -1 \ (mod \ p)
    Use Euler's criterion. Thus, -1 is a quadradic residue if and only if (-1)^{(p-1)/2} \equiv 1 (\bmod p) hence if p\equiv 1(\bmod 4) then (p-1)/2 is even and the congruence holds.
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