Proposition

The Legendre Symbol satisfies: 1. $\left( \frac{a}{p} \right)$ depends only on the residue of $a\pmod{p}$; 2. $$\left( \frac{ab}{p} \right)=\left( \frac{a}{p} \right)\left( \frac{b}{p} \right)$$ 3. For any $b$ coprime to $p$, we have $$\left( \frac{ab^{2}}{p} \right)=\left( \frac{a}{p} \right)$$ 4. $$\left( \frac{-1}{p} \right)=(-1)^{\frac{p-1}{2}}$$