Let a be in Z[i], show that if N(a) is prime in Z, then a is irreducible in Z[i].
Assume that $\displaystyle a = bc$ then $\displaystyle N(a) = N(b)N(c)$ if $\displaystyle N(a)$ is prime then one of $\displaystyle N(b)$ or $\displaystyle N(c)$ is $\displaystyle 1$ since the norm is $\displaystyle 1$ it means the element is a unit. And so $\displaystyle b$ or $\displaystyle c$ is a unit. Thus $\displaystyle a$ is irreducible.