Factoring ideal into prime ideals in $\mathbb{Z}[\sqrt{-5}]$

Question

I would like to write the ideal $(9)$ as a product of prime ideals in $\mathbb{Z}[\sqrt{-5}]$, which is a Dedekind domain. We have
$$9=3 \cdot 3=(2+\sqrt{-5})\cdot (2-\sqrt{-5}) $$
and I have shown that $3,2+\sqrt{-5},2-\sqrt{-5}$ are irreducible and not prime.
I can see that $(3)$ is not a prime ideal because $(2+\sqrt{-5}) (2-\sqrt{-5})=9 \in (3)$, but $2+\sqrt{-5},2-\sqrt{-5}\notin (3)$.
I am not sure what to do next.

Answer 1

This is exactly the situation that the Dedekind-Kummer theorem treats. This is given in the following two theorems from Keith Conrad's Factoring after Dedekind.

Theorem 1 (Dedekind). Let $K$ be a number field and $\alpha \in O_K$ such that $K= \mathbb{Q}(\alpha)$. Let $f(T)$ be the minimal polynomial of $\alpha$ in $\mathbb{Z}[T]$. For any prime $p$ not dividing $[O_K:\mathbb{Z}[\alpha]]$, write $$ f(T) \equiv \pi_1(T)^{e_1}\cdots \pi_g(T)^{e_g} \pmod{p} $$ where the $\pi_i(T)$ are distinct monic irreducibles in $\mathbb{F}_p[T]$. Then $ (p) = p O_K$ factors into prime ideals as $$ \newcommand{\p}{\mathfrak{p}} (p) = \p_1^{e_1} \cdots \p_g^{e_g} $$ where there is a bijection between the $\p_i$ and $\pi_i(T)$ such that $N(\p_i) = p^{\deg(\pi_i)}$. In particular, this applies for all $p$ if $O_K=\mathbb{Z}[\alpha]$.

He further describes how to obtain generators for the $\p_i$ in Theorem 8.

Theorem 8. In the notation of Theorem 1, when $\p_i$ is the prime ideal corresponding to $\pi_i(T)$ we have the formula $\p_i = (p,\Pi_i(\alpha))$ where $\Pi_i(T)$ is any polynomial in $\mathbb{Z}[T]$ that reduces mod $p$ to $\pi_i(T)$ mod $p$.

Returning to your problem, the minimal polynomial of $\sqrt{-5}$ is $T^2 + 5$. How does this factor mod $3$? What does this tell you about the primes ideals in the factorization of $(3)$ and their generators?