Nach Fakt ist unter Verwendung von Fakt und Fakt

${\displaystyle {}{\begin{aligned}\left({\frac {q}{p}}\right)&=\left({\frac {\left({\frac {-1}{p}}\right)p}{q}}\right)\\&=\left({\frac {\left({\frac {-1}{p}}\right)}{q}}\right)\cdot \left({\frac {p}{q}}\right)\\&=\left({\frac {(-1)^{\frac {p-1}{2}}}{q}}\right)\cdot \left({\frac {p}{q}}\right)\\&=\left({\frac {-1}{q}}\right)^{\frac {p-1}{2}}\cdot \left({\frac {p}{q}}\right)\\&={\left((-1)^{\frac {q-1}{2}}\right)}^{\frac {p-1}{2}}\cdot \left({\frac {p}{q}}\right)\\&=(-1)^{{\frac {q-1}{2}}\cdot {\frac {p-1}{2}}}\cdot \left({\frac {p}{q}}\right).\end{aligned}}}$