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Show that if p == 1 mod 4, then (a/p) = (-a/p).

(Note that == means congruent).

I know that if X^2==a mod p (p is a prime) is solvable then a is a quadratic residue of p.

For an example, I let p = 5 since 5==1 mod 4. Then, I let X = 2 and 4 just to check the equation. So:

2^2==-1 mod 5...........a=-1 is quadratic residue.

4^2==1 mod 5............a=1 is quadratic residue.

I know that the legendre symbol (a/p) is 1 if a is a quadratic residue mod p and -1 if a is a quadratic non-residue.

From my example, (-1/5)=1 and (1/5)=1, so I have found an example that shows (a/p) = (-a/p) but I'm having trouble proving it in general.

Thanks!

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# Quadratic congruences with prime modulus

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