Multiplicativity of the Legendre symbol

Statement

Let $p$ be an odd prime, and let $a$ and $b$ be integers. Then $$\left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right) \left(\frac{b}{p}\right),$$ where the symbols are Legendre symbols.

Remarks

The proof follows from Euler's criterion, because $(ab)^{(p-1)/2} = a^{(p-1)/2} b^{(p-1)/2}$. In words: modulo $p$, the product of two nonresidues is a residue, and the product of a residue and a nonresidue is a nonresidue. For example, $3$ and $5$ are nonresidues modulo $7$, and $15 \equiv 1 \pmod 7$ is a residue. So the map $a \mapsto \left(\frac{a}{p}\right)$ is a completely multiplicative function of $a$.

Explanations

L1 human

To decide whether a product is a square modulo $p$, you can test each factor. Two "no" answers give "yes", just as two negative signs give a positive sign.