Legendre Symbol is Multiplicative

Theorem
Let $p$ be a odd prime.

Let $a, b \in \Z$.

Then:


 * $\paren {\dfrac {a b} p} = \paren {\dfrac a p} \paren {\dfrac b p}$

where $\paren {\dfrac a p}$ is the Legendre symbol.

Proof
We have: