Find all maximal ideals of the ring $\mathbb Z[i]$ containing the ideal $I=(4-2i) \mathbb Z[i] + (9+3i) \mathbb Z[i]$

Question

Find all maximal ideals of the ring $\mathbb Z[i]$ containing the ideal $I=(4-2i) \mathbb Z[i] + (9+3i) \mathbb Z[i]$

I think that in this task I need to find gcd first: $$4-2i=2(2-i)=(1-i)(1+i)(2-i)$$ $$9+3i=3(3+i)=3(1+i)(2-i)$$ $$\text{gcd}(4-2i,9+3i) ∼ (1+i)(2-i)=3+i$$

However, I don't know what I can do next.

Answer 1

Hint: $\mathbb Z[i]$ is a PID. In a PID, to divide is to contain. Thus $(1+i)(2-i)=3+i$ gives $$ \langle 1+i \rangle \supseteq \langle 3+i \rangle, \quad \langle 2-i \rangle \supseteq \langle 3+i \rangle $$