Prove that $\left(\frac{-1}{p}\right)\left(\frac{3}{p}\right)=\left(\frac{p}{3}\right)$

Question

Why does $\left(\frac{-1}{p}\right)\left(\frac{3}{p}\right)$ equal $\left(\frac{p}{3}\right)$?

Answer 1

Use quadratic reciprocity :

$$ \left (\frac{p}{3} \right )=\left (\frac{3}{p} \right ) \cdot (-1)^{\frac{3-1}{2} \cdot \frac{p-1}{2}}$$

But we also know that : $$(-1)^{\frac{p-1}{2}}=\left (\frac{-1}{p} \right )$$ and thus the conclusion follows :

$$\left (\frac{p}{3} \right )=\left (\frac{3}{p} \right ) \cdot \left (\frac{-1}{p} \right )$$ by combining the two relations.