Let be $Z[\sqrt-3 ]=\lbrace a+b\sqrt-3: a,b \in Z\rbrace$. With the usual operations in the complex numbers. Prove that Z is an integral domain and that 2 is irreducible in $Z[\sqrt-3]$.
It's easy to prove that the set together with the operations is a ring with unity. I am not sure how to prove that it has no divisors of zero and that 2 is irreducible. The explicitation of those steps will be very usefull. Thank you.