Show that $1-i$ is irreducible in $\mathbb{Z}[i]$?

Question

I am using contemporary abstract algebra, by Joseph, I am in divisibility factor, can you please provide details step to solve above example with explanations.

Answer 1

Hint: The norm function $\mathbb{Z}[i] \to \mathbb{Z}$, given by $\|a + bi\| = a^2 + b^2$, satisfies $\|xy\| = \|x\| \cdot \|y\|$. So if $x$ divides $1 - i$, then $\|x\|$ divides $\|1 - i\|$. What is $\|1 - i\|$, and how many integers divide it? What possible $x$s have those norms?