Legendre symbol and from $\mathbb{Z}/p\mathbb{Z}$

Question

Let's define map $$\Phi:\mathbb{Z}/p\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}$$ as $$\Phi(x) = x^{(p-1)/2}.$$ It's easy to proof that $$\Phi(x) = 1,~~~\text{iff}~~~x = b^2~~~\text{for some}~~~b\in\mathbb{Z}/p\mathbb{Z}$$ and $$\Phi(0) = 0.$$ And consider Legendre symbol $$\left(\frac{n}{p}\right): \mathbb{Z}/p\mathbb{Z}\to \{\pm1, 0\}$$ Why it's false that $$\left(\frac{n}{p}\right) = \Phi(n)?$$ More exactly, that $$\left(\frac{n}{p}\right) = 1\in\mathbb{Z},~~~\text{if}~~~\Phi(n) = 1\in\mathbb{Z}/p\mathbb{Z},$$ $$\left(\frac{n}{p}\right) = -1\in\mathbb{Z},~~~\text{if}~~~\Phi(n) = - 1\in\mathbb{Z}/p\mathbb{Z}$$ and $$\left(\frac{n}{p}\right) = 0,~~~\text{if}~~~\Phi(n) = 0.$$ Thanks.