Legendre symbol multiplicativity

Is there some formal proof of Legendre symbol's being multiplicative? It seems pretty intuitive but I can't think of a way to formally show it?

• You have to show that if $a$ and $b$ are not a square then $ab$ is a square. Once you know $(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic it becomes obvious. – reuns Oct 14 '16 at 5:20