<s>[INST] Proving $2,3,1+\sqrt{-5}$ and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$ Could anyone help me prove that $2,3,1+\sqrt{-5}$ and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$?
As $6=2*3=(1+\sqrt{-5})(1-\sqrt{-5})$ so $\mathbb{Z}[\sqrt{-5}]$ is not a UFD. Therefore is not a PID or euclidean domain
 [/INST] The standard method is:
Define a function $N\colon \mathbb{Z}[\sqrt{-5}]\to\mathbb{Z}$ by $N(a+b\sqrt{-5}) = (a+b\sqrt{-5})(a-b\sqrt{-5}) = a^2+5b^2$.


*

*Prove that $N(\alpha\beta) = N(\alpha)N(\beta)$ for all $\alpha,\beta\in\mathbb{Z}[\sqrt{-5}]$.

*Conclude that if $\alpha|\beta$ in $\mathbb{Z}[\sqrt{-5}]$, then $N(\alpha)|N(\beta)$ in $\mathbb{Z}$.

*Prove that $\alpha\in\mathbb{Z}[\sqrt{-5}]$ is a unit if and only if $N(\alpha)=1$.

*Show that there are no elements in $\mathbb{Z}[\sqrt{-5}]$ with $N(\alpha)=2$ or $N(\alpha)=3$.

*Conclude that $2$, $3$, $1+\sqrt{-5}$, and $1-\sqrt{-5}$ are irreducible.


This is a common technique for dealing with rings of the form $\mathbb{Z}[\theta]$, where $\theta$ is an algebraic integer.
It can also be done directly, though it is a bit more laborious. Here's what I came up with on the fly:
Assume that $(a+b\sqrt{-5})(c+d\sqrt{-5}) = 2$. Then $ac+5bd = 2$ and $ad+bc=0$. If $a=0$, then we must have $c=0$ (since $bc=0$ but we cannot have $a=b=0$), but then $5bd=2$ is impossible. Thus, $a\neq0$ and $c\neq 0$. If $b=0$, then $d=0$, so the factorization occurs in $\mathbb{Z}$ and is trivial; symmetrically if $d=0$. So we may assume that all of $a,b,c,d$ are nonzero.
Then $ad=-bc$, so $2d=acd + 5bd^2 = -bc^2 + 5bd^2 = b(5d^2 - c^2)$. If $b$ is odd, then $b|d$, so writing $d=bk$ we get $abk + bc=0$, so $ak+c=0$, thus $c=-ak$. Hence $c+d\sqrt{-5} = -ak+bk\sqrt{-5} = k(-a+b\sqrt{-5})$. But this gives
$$2 = (a+b\sqrt{-5})(c+d\sqrt{-5}) = k(a+b\sqrt{-5})(-a+b\sqrt{-5}) = -k(a^2+5b^2).$$
Since $a$ and $b$ are both nonzero, $a^2+5b^2$ is at least 6, which is impossible. So $b$ is even, $b=2b'$. 
Then $b'(5d^2-c^2) = d$, so setting $5d^2-c^2 = k$ we have 
$$a+b\sqrt{-5} = a+2b'\sqrt{-5},\qquad c+d\sqrt{-5} = c+kb'\sqrt{-5}.$$
From $ad=-bc$ we get $ak=-2c$. If $a$ is even, then we have $a=2a'$, so
$$2 = (2a'+2b'\sqrt{-5})(c+d\sqrt{-5}) = 2(a'+b'\sqrt{-5})(c+d\sqrt{-5}),$$
which yields that $c+d\sqrt{-5}$ is a unit (in fact, this is impossible with $c$ and $d$ both nonzero, but that doesn't matter). If $a$ is odd, then $k$ is even and $c=ak'$, with $2k'=k$. So now we have
$$\begin{align*}
2 &= (a+b\sqrt{-5})(c+d\sqrt{-5})\\
&= (a + 2b'\sqrt{-5})(ak' + 2b'k'\sqrt{-5})\\
&= k'(a+2b'\sqrt{-5})(a+2b'\sqrt{-5})\\
&= k'(a^2 - 20b'^2) + 4ab'\sqrt{-5}
\end{align*}$$
which implies $a=0$ or $b'=0$ (hence $b=0$), contradicting our hypotheses.
Thus, the only factorizations of $2$ in $\mathbb{Z}[\sqrt{-5}]$ are trivial.
(And you can probably see why the "standard method" is so much better....)
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