I have the ring $\mathbb{Z}[\sqrt{-5}]$ and the function $\phi: \mathbb{Z}[\sqrt{-5}] \to \mathbb{Z}/3\mathbb{Z}$. I found the kernel of the function, namely $(3, 1-\sqrt{-5})$. Now I am trying to prove that the kernel is not a principal ideal. How can I do this?
I know I could use norms to do it. But what do I need to compute to conclude that this ideal is not a principal ideal?
The norm of a putative generator $\alpha$ must divide that of $3$, which is $9$ and that of $1-\sqrt{-5}$, namely $6$. Therefore $N\alpha=1$ or $3$.
If $N\alpha=1$ then $\alpha=\pm1$. That's impossible as $\phi(\pm1)\ne0$.
But $N\alpha=3$ is impossible: there are no integers with $a^2+5b^2=3$.