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Using one of the reciprocity laws evaluate the quadratic Gauss sum $G(2;p)$. Comparing with the formula $G(2;p)=(2|p)G(1;p)$ deduce that $(2|p)=(-1)^{(p^2-1)/8}$ if $p$ an odd prime.

From Apostol Chapter 9. I can't see what is meant by the initial hint of using one of the reciprocity laws.

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up vote 1 down vote accepted

This is a two-part question - I'll ignore the second part. The first part is "Using one of the reciprocity laws evaluate the quadratic Gauss sum G(2;p)." I believe the reciprocity law you want to find is Theorem 9.16 involving the function S(a,m). You should get

$ G(2;m)=S(4,m)=\sqrt{\frac p 4} \frac{1+i}{\sqrt 2} \overline{S(m,4)} $

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What is $S(a,m)$? –  JavaMan Nov 23 '11 at 17:41
    
@Colin Thanks that works. –  apatch Nov 24 '11 at 15:03
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