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Thread: Quad Residues

  1. #1
    Junior Member
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    Quad Residues

    Question:

    Prove: If p = 1(mod 4) is a prime, then -4 and (p-1)/4 are both quadratic residues of p.
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  2. #2
    Super Member PaulRS's Avatar
    Joined
    Oct 2007
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    Since 4 is a perfect square: $\displaystyle \left( {\tfrac{{ - 4}}
    {p}} \right) = \left( {\tfrac{{ - 1}}
    {p}} \right)\left( {\tfrac{4}
    {p}} \right) = \left( {\tfrac{{ - 1}}
    {p}} \right) = {\left( { - 1} \right)^{\tfrac{{p - 1}}
    {2}}} = 1$

    Let's denote $\displaystyle a=\left( {\tfrac{{p - 1}}
    {4}} \right)$. We have $\displaystyle \left( {\tfrac{a}
    {p}} \right) = \left( {\tfrac{a}
    {p}} \right)\left( {\tfrac{{ - 4}}
    {p}} \right) = \left( {\tfrac{{ - 4a}}
    {p}} \right) = \left( {\tfrac{{ - \left( {p - 1} \right)}}
    {p}} \right) = \left( {\tfrac{1}
    {p}} \right) = 1$
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