Past exam of the mathematics course of the University of Cambridge / 2003 / ib / Paper 1 / 17G / b / Solution

Apply Euler's criterion to $-1$. It givesFor $p\nmid ab$, the same criterion gives $(ab/p)\equiv (ab{)}^{(p-1)/2}=a^{(p-1)/2}{b}^{(p-1)/2}\equiv (a/p)(b/p)(\mathrm{mod}p)$. Both sides are integers in $\{1,-1\}$, which are distinct modulo an odd prime, so congruence implies equality. This proves multiplicativity of the Legendre symbol: