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Math Help - Quadratic Reciprocity/ Euler's Criterion

  1. #1
    Junior Member
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    Quadratic Reciprocity/ Euler's Criterion

    Question:
    If ab = r (modp), where r is a quadratic residue of the odd prime p, prove that a and b are both quadratic residues of p or both non-residues of p.
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  2. #2
    Super Member PaulRS's Avatar
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    Remember that <br />
x \equiv y\left( {\bmod .p} \right) <br />
implies \left( {\tfrac{x}<br />
{p}} \right) = \left( {\tfrac{y}<br />
{p}} \right)<br />
so that 1=\left( {\tfrac{{r}}<br />
{p}} \right)= \left( {\tfrac{{ab}}<br />
{p}} \right)=<br />
\left( {\tfrac{a}<br />
{p}} \right) \cdot \left( {\tfrac{b}<br />
{p}} \right) <br />
now multiple by <br />
\left( {\tfrac{a}<br />
{p}} \right)<br />
and we have: <br />
\left( {\tfrac{a}<br />
{p}} \right) = \underbrace {\left( {\tfrac{a}<br />
{p}} \right)^2 }_{ = 1} \cdot \left( {\tfrac{b}<br />
{p}} \right)<br />
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