Show that $2,3, 1-\sqrt{-5}, 1+\sqrt{-5}$ are irreducible over $\mathbb{Z}[\sqrt{-5}]$, but not prime and that 1 and -1 are the only units.
Let $N$ be the norm map into $\mathbb{Z}$ and let u denote the unit, then because it is a homomorphism it follows that :
$N(xy)=N(x)N(y)$ and thus with $N(u)N(u^{-1})=N(uu^{-1})=N(1)$ it follows that the norm of u is a divider of 1 and thus also a unit of $\mathbb{Z}$. Therefore we have in $\mathbb{Z}[\sqrt{-5}]$: $\forall a,b \in \mathbb{Z}: N(a+b\sqrt{-5})=(a+b\sqrt{-5})(a-b\sqrt{-5})=a^{2}+5b^{2}=1 \ \ \Rightarrow u= \pm 1$
Now it is to show that $2,3,1-\sqrt{-5}, 1+\sqrt{-5}$ are irreducible over $\mathbb{Z}[\sqrt{-5}]: $
Assume $1-\sqrt{-5}$ is reducible, then there must exist $a,b \in \mathbb{Z}[\sqrt{-5}]$ so that $N(1-\sqrt{-5})=N(a)N(b) \Rightarrow N(a)=N(b)= \pm (1-\sqrt{-5})$ But since $1-\sqrt{-5}$ is not a quadratic remainder of $5$, there doesn't exist a solution for the equations $a^{2}+5b^{2}= \pm (1-\sqrt{-5})$ And the exactly same argument also works for $2,3$ and $1+\sqrt{-5}$.
Thus we have shown that $2,3,1\pm \sqrt{-5}$ are not reducible over $\mathbb{Z}[\sqrt{-5}]$.
Now we show that they are not prime:
Assume that 2 is a prime in $\mathbb{Z}[\sqrt{-5}]$, then because of $2\cdot 3 = (1-\sqrt{-5})(1+\sqrt{-5})=6$ it must hold that $2|(1-\sqrt{-5})$ or $2|(1+\sqrt{-5})$. But with $a,b \in \mathbb{Z}[\sqrt{-5}]$ it immediately follows that for :
$(1\pm\sqrt{-5})= 2(a+b\sqrt{-5})$ $2b = \pm 1$. So 2 is not a prime in $\mathbb{Z}[\sqrt{-5}]$.
Assume that 3 is a prime in $\mathbb{Z}[\sqrt{-5}$, then: $(1\pm \sqrt{-5}) = 3(a+b\sqrt{-5}) \Rightarrow 3b= \pm 1$ it follows that 3 is not a prime in $\mathbb{Z}[\sqrt{-5}]$.
Assume that $1\pm \sqrt{-5}$ is a prime in $\mathbb{Z}[\sqrt{-5}]$, then: $3 = (1\pm \sqrt{-5})(a+b\sqrt{-5}) \Rightarrow (1\pm \sqrt{-5} )a = 3$ which is not solvable in $\mathbb{Z}$. And also $2=(1\pm \sqrt{-5})(a+b\sqrt{-5})=(1\pm \sqrt{-5})a$ which is also not solvable in $\mathbb{Z}$ and thus $(1\pm \sqrt{-5})$ can not be a prime in $\mathbb{Z}[\sqrt{-5}]$.
Tell me if this proof is correct. Please.