Find all the ideals of $\mathbb { Z } [ i ]$ containing $(5)$

Question

Problem : Find all the ideals of $\mathbb { Z } [ i ]$ containing $(5)$, the principal ideal generated by $5$.

I already know that $\mathbb { Z } [ i ]$ is isomorphic to $\mathbb{ Z }[x]/(x^2+1)$ but I have troubles getting an intuitive interpretation of polynomial rings and quotient rings. Should I be looking for principal ideals generated by polynomial with $5$ as a root?

Answer 1

Hint: The set of ideals of $\Bbb{Z}[i]$ containing $(5)$ corresponds bijectively to the set of ideals of $\Bbb{Z}[i]/(5)$.