While proving that $\langle 3,2+\sqrt{-5}\rangle$ is not a principal ideal of $\mathbb{Z}[\sqrt{-5}]$, the norm function $N(a+b\sqrt{-5})=a^2+5b^2$ has been used to arrive at a contradiction. This in turn proves that $\mathbb{Z}[\sqrt{-5}]$ is not a euclidean domain.
My question is that we get a norm function only when the ID is an ED. Then how can we consider this norm function and then arrive at our contradiction because since this ID is not an ED such a norm function must not exist. And even if this norm can be considered then how can we draw our conclusion that $\mathbb{Z}[\sqrt{-5}]$ is not ED just from this norm because norm function is not unique. So there may exist some other norm function for which this contradiction will not arise.
Also how can I find the norm function for an ED in order to prove that an ID is an ED in other questions where it is asked to prove that an ID $R$ is an ED ?
I am stuck here very badly and I am just a beginner in this topic. That's why my concepts are not clear. I am having a lot of doubts regarding ED. Any explanation regarding this question and about any other concepts which you might think will be useful about ED will be extremely helpful. Thanking in advance.
I think there is a bit of confusion. Any ring of integers $\mathcal{O}$ has a norm function $N:\mathcal{O}\to \mathbb{Z}$. It has nothing to do a priori with the fact that $\mathcal{O}$ is, or is not, euclidean.
You can use this norm function in any proof, for instance when proving that $\mathbb{Z}[\sqrt{-5}]$ is not principal, which proves that it is not euclidean with respect to any function $\mathcal{O}\to \mathbb{Z}$.
A given ring of integers might not be euclidean at all, it might be euclidean with respect to its usual "algebraic" norm $N:\mathcal{O}\to \mathbb{Z}$, or it might be euclidean with respect to some other function $\mathcal{O}\to \mathbb{Z}$. Anything is possible.