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Math Help - Legendre symbol proof

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    Legendre symbol proof

    Prove that if p>3 is an odd prime, then (-3/p) = 1 if p==1(mod 6) and -1 if p==5(mod 6). Note that (-3/p) is the Legendre symbol, not division. Thank you.
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    MHF Contributor Bruno J.'s Avatar
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    It's a direct application of quadratic reciprocity!
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    Can you get me started on the proof? I know that the Law of Quadratic Reciprocity states that if p and q are odd primes and p==q==3(mod 4), then (p/q) = -(q/p). Otherwise, (p/q) = (q/p). But I am not seeing an instant connection to mod 6...
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    MHF Contributor Bruno J.'s Avatar
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    Another way of writing the law of quadratic reciprocity is

    \left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{(p-1)(q-1)/4}.

    See what you get from there!
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