Demonstrating that a particular ideal is maximal in a number field I have three ideals, each with two elements of $\mathbb Z[\sqrt{-5}]$. If you show me how to show one of them is maximal (hence prime) then I think I can manage to do the remaining two by myself.
$\langle 2, 1 + \sqrt{-5}\rangle$

We have 
$$\begin{eqnarray*} \Bbb{Z}[\sqrt{-5}]/(2,1+ \sqrt{-5}) &\cong& \Bbb{Z}[x]/(x^2 + 1)/\left((2,1+x)/(x^2 + 1)\right) \\
&\cong& \Bbb{Z}[x]/(2,x+1) \\
&\cong& \Bbb{Z}/2\Bbb{Z}\end{eqnarray*}$$
and so your ideal is maximal.
