$Φ_n$ is Euler group, $n> 2$ is an integer, and $m$ the number of solutions of the equation $x^2 = 1$ in the ring $Z_n$.

Question

• Prove

$$\prod_{i ∈ Φ_n} i=(-1)^{\frac{m}{2}}$$

• Then what becomes this identity if $n$ is a prime number?

I know that if $x^2=1$, we pair the number with its inverse modulo $n$ in the product. Otherwise, we pair it up with $−x$. But still don't know how to prove it.

Can please anyone help me with this proof?