The question is: 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!