Let $\mathbb{Z}[\frac{{1+\sqrt{-3}}}{2}]$ be the ring of integers where $\alpha=\frac{{1+\sqrt{-3}}}{2}$. I wish to show that $1-\alpha$ is a prime and check whether 3 is a prime or not.
To show that $1-\alpha$ is prime, suppose $1-\alpha$ divides $(a+b\alpha)(c+d\alpha)$. Then I wish to show that it divides one of them. I wrote down the division algorithm for $1-\alpha$ divides $(a+b\alpha)(c+d\alpha)$ but it's a big mess that I can't seem to make sense of. Any help is appreciated.