Show that (2) is prime ideal in $\mathbb{Z}[i]$ I want to show that (2) is a prime ideal in $\mathbb{Z}[i]$. I know that I need to show that the quotient ring $\mathbb{Z}[i]/2$ is an integral domain. I'm not sure how this can be done or what $\mathbb{Z}[i]/2$ looks like. I also know that $\mathbb Z[i]\simeq \mathbb Z[x]/(x^2+1)$, but I'm not sure why.
 A: 
I want to show that (2) is a prime ideal in $\mathbb{Z}[i]$.

Did you consider $2=(1+i)(1-i)$?

I'm not sure how this can be done or what $\mathbb{Z}[i]/2$ looks like.

Maybe you should read a couple other solutions on the site about understanding quotient rings, like this one

I also know that $\mathbb Z[i]\simeq \mathbb Z[x]/(x^2+1)$, but I'm not sure why.

You can prove that using the first homomorphism theorem for rings with a map from $\mathbb Z[x]\to \mathbb Z[i]$ given by substituting $i$ into polynomials in $\mathbb Z[x]$.
A: This claim is false. Observe that $0 = (x+1)(x+1) \in \mathbb{Z}[x]/(x^2+1,2)$. Conclude that the quotient is not a domain, hence that (2) is not prime.
A: I don't think $(2)$ is a prime ideal of $\mathbb{Z}[i]$. I hope that this can answer your second question: The map $\varphi: \mathbb{Z}[X] \to \mathbb{Z}[i], f \mapsto f(i)$ is a well defined ring homomorphism. One can now check that $\text{ker}(\varphi)=(x^2+1)$ and that $\varphi$ is indeed surjective, hence $\mathbb{Z}[X]/(x^2+1) \cong \mathbb{Z}[i]$.
A: Yes, $\,R = \Bbb Z[i]\cong \Bbb Z[x]/f,\ f =x^2\!+\!1,\,$ follows immediately from the First Isomorphism Theorem, as in adh's answer. Since you seek to  prove that $2$ is not prime in $\,R\,$ by showing $\,R/2\,$ is not a domain, let's do that by using  quotient reciprocity as follows
$$\begin{align} \Bbb Z[i]/2 &\,\cong\,  (\Bbb Z[x]/f)\,/\,(2,f)/f\\[.2em] 
&\,\cong\, \ \Bbb Z[x]/(2,f)\\[.2em]
&\,\cong\, (\Bbb Z[x]/2)\,/\,(f,2)/2\\[.2em]
&\,\cong\ \ \Bbb F_2[x]/f \\[.2em]
&\,\cong \ \ \Bbb F_2[x]/(x\!+\!1)^2
\end{align}\qquad$$
The final quotient is not a domain since there $\,x\!+\!1 \neq 0$ but $\,(x\!+\!1)^2 = 0.\,$ This is called algebra of dual numbers over $\Bbb F_2.\,$ Dual numbers are useful as  algebraic models of tangent and jet spaces.
