I am stuck in reading Marcus book, Chapter 3.
Let $D=\mathbb Z[\sqrt{-5}]$.
I need to verify that $$(1-\sqrt{-5})=(2,1+\sqrt{-5})(3,1-\sqrt{-5}).$$
I proved that $(1+\sqrt{-5})=(2,1+\sqrt{-5})(3,1+\sqrt{-5})$.
According to Marcus book, $(2,1+\sqrt{-5}),(3,1+\sqrt{-5}),(3,1-\sqrt{-5})$ are prime ideals. Why?
I know the target is to prove the factorization of ideal $(6)$ is unique of prime ideals.
I'll do $\langle2,1+\sqrt 5 \rangle$. You'll get the idea.
The idea is that the quotient by a prime ideal should yield a domain:
So we have the ring $D=\mathbb Z[\sqrt 5]$ which I'm going to describe as $\mathbb Z[x]/\langle x^2-5\rangle$. We are further quotienting by the ideal $I=\langle 2, 1+x \rangle$ in this ring. Noticing that $x^2-5 = (x-1)(x+1)-2\cdot2\in\langle 2, 1+x \rangle$, we have
$$D/I\cong \mathbb Z[x]/\langle 2,1+x\rangle/\langle x^2-5 \rangle \cong \mathbb Z[x]/\langle 2, x+1 \rangle \cong \mathbb Z/2\mathbb Z,$$
which is a domain.