I'm doing Exercise 9 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.
Show that every ideal in the ring of Gaussian integers is principal.
Could you please verify if my attempt is fine or contains logical mistakes? Thank you so much for your help!
Let
$\mathbb Z[i]$ denote the ring of Gaussian integers.
The norm $N(z) = z \overline{z}$ for $z \in \mathbb Z[i]$.
$I$ be an ideal of $\mathbb Z[i]$.
Among all non-zero elements of $I$, we pick $a$ such that $N(a)$ is the smallest. For $b \in I$, by Euclidean division algorithm, there are $r,s \in \mathbb Z[i]$ such that $b =ra+s$ and $N(s) \le N(a)/2$. By construction, $s=0$ and thus $I = \langle a \rangle$. This completes the proof.