Prove that $(3, 2 + \sqrt{-5})$ is maximal ideal in $\mathbb Z[\sqrt{-5}]$ Prove that $(3, 2 + \sqrt{-5})$ is prime ideal in $\mathbb Z[\sqrt{-5}]$. I want to show that $(3, 2 + \sqrt{-5})$ is maximal directly from definition, ie. show that by adding any other element to it we get the entire ring. Any hints on how to do this? I already know that the fact can be shown by using different isomorphisms.
 A: Consider $(3,2+\sqrt{-5},a+b\sqrt{-5})$. You have $$(3,2+\sqrt{-5},a+b\sqrt{-5})=(3,2+\sqrt{-5},a+b\sqrt{-5}-b(2+\sqrt{-5}))=\\=(3,2+\sqrt{-5},a-2b)=\begin{cases}(3,2+\sqrt{-5})&\text{if }3\mid a-2b\text{ in }\Bbb Z\\ (1)&\text{if }3\nmid a-2b\text{ in }\Bbb Z\end{cases}$$
The first case occurs when $a+b\sqrt{-5}\in(3,2+\sqrt{-5})$. 
A: Hint $ $ Write $\, w = \sqrt{-5}\ $ and employ $\rm\color{#0a0}{modular}$ reduction as in Euclid's GCD algorithm
$$\begin{align} (w\!+\!2,\,3,\, a\!+\!bw) 
&=\, (w\!+\!2,\,3,\: a\!+\!\color{#c00}w\,b\color{#0a0}{\bmod}{w\! +\! 2})\\
&=\, (w\!+\!2,\,3,\, a\!\color{#c00}{-2}\,b)\ \ {\rm by}\ \ \color{#c00}w\equiv\color{#c00}{-2}\!\!\!\pmod{\!w\!+\!2}\\
&=\, (w\!+\!2,\,3)\ \ \,{\rm if}\ \ (3,\,a\!-\!2b)=3\ \ {\rm else}\\ 
&=\, (1)\qquad\quad \ \, {\rm if}\ \ (3,\,a\!-\!2b)=1
\end{align} $$
Hence $\ a\!+\!bw\not\in I = (3,w\!+\!2)\,\Rightarrow\, (I,a\!+\!bw) = (1),\ $ so $\,I\,$ is maximal
Remark $\ $ The above modular reduction is essentially a special case of normal form reduction modulo a Hermite normal form basis - a powerful method which proves very useful for computation. See this answer for more.
A: $\mathbb Z[\sqrt{-5}]\cong \mathbb Z[x]/(x^2+5)$, then $\mathbb Z[\sqrt{-5}]/(3,2+\sqrt{-5})\cong  \mathbb Z[x]/(x^2+5,3,2+x)\cong \mathbb Z_3[x]/(x^2+5,2+x)=\mathbb Z_3[x]/(x^2+2,x+2)$
In other hand we have in $\mathbb Z_3$ that $x^2+2=(x+2)(x+1)$, so $x^2+2\in(x+2)$ and $(x^2+2,x+2)=(x+2)$, so $\mathbb Z_3[x]/(x^2+2,x+2)=\mathbb Z_3[x]/(x+2)\cong \mathbb Z_3[-2]=\mathbb Z_3[1]=\mathbb Z_3$, then $\mathbb Z[\sqrt{-5}]/(3,2+\sqrt{-5})\cong \mathbb Z_3$ and $(3,2+\sqrt{-5})$ is maximal ideal.
