 Solution

Euler's criterion states, for an odd prime $p$,Apply it to $a=-1$. The integer $(-1{)}^{(p-1)/2}$ is $1$ when $p\equiv 1(\mathrm{mod}4)$ and $-1$ when $p\equiv 3(\mathrm{mod}4)$. Since those signs are distinct modulo an odd prime, the modular congruence determines the actual Legendre symbol:This is the first supplementary law for quadratic reciprocity.