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Math Help - bout the quadratic reciprocity law

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by felixmcgrady View Post
    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 r is primitive root then \{r^2,r^4,...,r^{p-1}\} are the quadradic residues i.e. the even exponents of r. Their product is r^{2+4+...+(p-1)}. Notice that 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 r^{(p^2-1)/4}.
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