Legendre symbol and Euler's criterion

Definition. For $p$ odd prime and $a\in\mathbb{Z}$:
$$\left(\frac{a}{p}\right)=\begin{cases}1&\text{if }a\text{ is a QR mod }p,\ a\not\equiv0,\\ -1&\text{if }a\text{ is a QNR mod }p,\\0&\text{if }p\mid a.\end{cases}$$

Euler's criterion. $\left(\frac{a}{p}\right)\equiv a^{(p-1)/2}\pmod p$.

Multiplicativity. $\left(\frac{ab}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)$.

Conclusion. The Legendre symbol is a multiplicative homomorphism from $\mathbb{F}_p^*$ to $\{\pm1\}$.