Integral domain (rings and fields)

Question

Let $\mathbb Z[i]=\{a+ib \mid a, b \in \mathbb Z \}$. How to Show that $\mathbb Z[i]$ is a integral domain?

Answer 1

Assuming that you gave shown it to be a ring, then you need to show that there are no zero divisors. What it means is to show that no two non-zero complex numbers can be multiplied to produce $0$.

Answer 2

$$ \mathbb{Z}[i] \cong \mathbb{Z}[x]/(x^2+1) $$ but $x^2 + 1 $ is irreducible in the UFD $\mathbb{Z}[x]$ so the ideal $ (x^2 + 1 )$ is prime and thus $ \mathbb{Z}[i] $ is a domain.