# Thread: bout the quadratic reciprocity law

1. ## bout the quadratic reciprocity law

Hey can any body help me with these?

1. if r is a primitive root of the odd prime p, prove that the product of the quadratic residues of p is congruent modulo p to r^((p^2-1)/4) and the product of the nonresidues of p is congruent modulo p to r^((p^2-1)/4).

2. if the prime p>3, show that p divides the sum of its quadratic residues.

3. if the prime p>5, show that p divides the sum of the squares of its quadratic nonresidues.

Thanks very much, i just dont get these at all.

2. Originally Posted by felixmcgrady
1. if r is a primitive root of the odd prime p, prove that the product of the quadratic residues of p is congruent modulo p to r^((p^2-1)/4) and the product of the nonresidues of p is congruent modulo p to r^((p^2-1)/4).
I do this one and #2 and #3 should be similar. If $\displaystyle r$ is primitive root then $\displaystyle \{r^2,r^4,...,r^{p-1}\}$ are the quadradic residues i.e. the even exponents of $\displaystyle r$. Their product is $\displaystyle r^{2+4+...+(p-1)}$. Notice that $\displaystyle 2+4+...+(p-1) = 2\left( 1 + 2 + ... + \tfrac{p-1}{2} \right) = \left( \tfrac{p-1}{2} \right) \left( \tfrac{p+1}{2} \right) = \tfrac{p^2-1}{4}$.
Thus, their product is congruent to $\displaystyle r^{(p^2-1)/4}$.