Solutions to the equation $x^2=1$ in a cyclic group

Question

In a cyclic group, say $(\mathbb{Z}/p\mathbb{Z})^*$ where $p$ is a prime, why does the equation $x^2=1 \mod p$ only have two solutions?

Thanks!

Answer 1

Because a quadratic equation in a field (more generally in an integral domain) has at most two roots. This is because $\alpha$ is a root of $p(x) \iff p(x) $ is divisble by $\;x-\alpha\;$ and $\;\deg p(x)q(x)=\deg p(x)+\deg q(x)$.