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Thread: 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 $\displaystyle
    x \equiv y\left( {\bmod .p} \right)
    $ implies $\displaystyle \left( {\tfrac{x}
    {p}} \right) = \left( {\tfrac{y}
    {p}} \right)
    $ so that $\displaystyle 1=\left( {\tfrac{{r}}
    {p}} \right)= \left( {\tfrac{{ab}}
    {p}} \right)=
    \left( {\tfrac{a}
    {p}} \right) \cdot \left( {\tfrac{b}
    {p}} \right)
    $ now multiple by $\displaystyle
    \left( {\tfrac{a}
    {p}} \right)
    $ and we have: $\displaystyle
    \left( {\tfrac{a}
    {p}} \right) = \underbrace {\left( {\tfrac{a}
    {p}} \right)^2 }_{ = 1} \cdot \left( {\tfrac{b}
    {p}} \right)
    $
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