I found a proof of quadratic reciprocity in wikipedia, which I don't quite understand. The link is http://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity.
On the last line of the Cyclotomic field setup part, it says $\left(\frac{q}{p}\right)=1$ iff $\sigma_q$ is an element of $H$. I only know $H$ is a subgroup of $\operatorname{Gal}(L/Q)$ of order $\frac{p-1}{2}$. But how could I know $\sigma_q$ is in $H$?