Show that $q$ must divide one of the prime integer factors of $N(q)$.

Question

The following is from Aluffi's Algebra

Why this is true? :

.. since q is prime in Z[i] ⊇ Z, q must divide one of the prime integer factors of N(q).

Answer 1

This might make a lot more sense to you if you look at specific numbers. I suggest $2$ and $3$.

Although $2$ is prime in $\mathbb{Z}$, it's not prime in $\mathbb{Z}[i]$, since $(1 - i)(1 + i) = 2$. So $N(2) = 4$, no surprise there. We also see that $N(1 - i) = 2$, and $N(1 + i) = 2$ as well. These facts may not seem obvious but they are easily verified.

And these facts mesh perfectly well with the multiplicative property of the norm function. If $ab = 2$, then $N(a)$ can only be $1$, $2$ or $4$, and the choice of $b$ follows immediately.

Compare $3$, which is indeed prime in both $\mathbb{Z}$ and $\mathbb{Z}[i]$. Its norm in the latter is still its square, and since it neither ramifies nor splits in this domain, it likewise follows that $N(z) = 3$ is impossible for $z \in \mathbb{Z}[i]$.