Suppose $d$ be a square-free integer and $\mathbb{Z}[\sqrt{d}]$ is a unique factorization domain.
If $p$ is prime and $d$ is a square mod $p$, then prove that $p$ is reducible in $\mathbb{Z}[\sqrt{d}]$.
Assume that $p$ is prime and $d$ is a square mod $p$.
By this assumption I can only get that $d$ is quadratic residue of $p$.
But, how to show that $p = (u_1 + v_1\sqrt{d})(u_2 + v_2\sqrt{d})$ in somehow ?