Quadratic Residues

**mndi1105** · November 3rd 2008, 05:34 PM

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.

**ThePerfectHacker** · November 3rd 2008, 07:15 PM

Originally Posted by mndi1105

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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