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Math Help - Quad Residues

  1. #1
    Junior Member
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    May 2009
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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: \left( {\tfrac{{ - 4}}<br />
{p}} \right) = \left( {\tfrac{{ - 1}}<br />
{p}} \right)\left( {\tfrac{4}<br />
{p}} \right) = \left( {\tfrac{{ - 1}}<br />
{p}} \right) = {\left( { - 1} \right)^{\tfrac{{p - 1}}<br />
{2}}} = 1

    Let's denote a=\left( {\tfrac{{p - 1}}<br />
{4}} \right). We have \left( {\tfrac{a}<br />
{p}} \right) = \left( {\tfrac{a}<br />
{p}} \right)\left( {\tfrac{{ - 4}}<br />
{p}} \right) = \left( {\tfrac{{ - 4a}}<br />
{p}} \right) = \left( {\tfrac{{ - \left( {p - 1} \right)}}<br />
{p}} \right) = \left( {\tfrac{1}<br />
{p}} \right) = 1
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