Showing that an ideal is not principal in $\mathbb{Z}[\sqrt{-21}]$

Question

I am trying to show that the ideal $$(2,\sqrt{-21}-1)(3, \sqrt{-21}) $$ is not principal in $\mathbb{Z}[\sqrt{-21}]$. Can anyone help with this?

Answer 1

The Very Useful Theorem that you want to use is that if an ideal $\mathfrak a$ is principal, equal to $\langle b\rangle$, then the norm of $\mathfrak a$ is $\mathbf N(b)$, where $\mathbf N$ is the field-theoretic Norm map from the big field down to $\Bbb Q$.

In your case, since an integral basis for the integers of $\Bbb Q(\sqrt{-21}\,)$ can be taken to be $\{1,\sqrt{-21}\,\}$ (because $-21\not\equiv1\pmod4$ ), so the Norm of the general integer $m+n\sqrt{-21}$ is $m^2+21n^2$. I’m sure you can finish it all off from here.