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Math Help - Quadratic Residues

  1. #1
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    Quadratic Residues

    Let p be a prime number with p = 1 mod 4. Prove that

    sum(from a=1 to (p-1)/2) of (a over p) =0 where (a over p) denotes the legendre symbol.
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  2. #2
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    Quote Originally Posted by mndi1105 View Post
    Let p be a prime number with p = 1 mod 4. Prove that

    sum(from a=1 to (p-1)/2) of (a over p) =0 where (a over p) denotes the legendre symbol.
    We know that \sum_{t=1}^{p-1}(t/p) = 0.

    Therefore, \sum_{t=1}^{(p-1)/2} (t/p) + \sum_{t=(p+1)/2}^{p-1} (t/p) = 0 .... [1]

    However, \sum_{t=(p+1)/2}^{p-1} (t/p) = \sum_{t=(p+1)/2}^{p-1} (-t/p) = \sum_{t=(p+1)/2}^{p-1} ((p-t)/p) = \sum_{t=1}^{(p-1)/2}(t/p)

    Therefore from [1] we get,
    2\sum_{t=1}^{(p-1)/2} (t/p) = 0 \implies \sum_{t=1}^{(p-1)/2}(t/p) = 0
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