Is the ideal $\{2m + (1 + \sqrt{-6})n:m, n\in\mathbb{Z}\}$ principal in $\mathbb{Z}[\sqrt{-6}]$?

Question

Is the ideal $\{2m + (1 + \sqrt{-6})n:m, n\in\mathbb{Z}\}$ principal in $\mathbb{Z}[\sqrt{-6}]$?

I have an exercise that asks just that. As a hint it says to prove that this ideal contains $1$, hinting that the authors consider it being principal correct answer. But it doesn't look to me like it's principal at all. So is it or is it not? Thanks.

Answer 1

What you've written isn't an ideal. What you presumably meant to write is $$I=(2,1+\sqrt{-6})=\{2\alpha+(1+\sqrt{-6})\beta:\alpha,\beta\in\mathbb{Z}[\sqrt{-6}]\}$$

Now, how to prove that $I$ is principal? I'll give a further hint there. Because $1+\sqrt{-6}\in I$ and $1-\sqrt{-6}\in\mathbb{Z}[\sqrt{-6}]$, we have (by the definition of ideal) $$7=1^2-(-6)=(1+\sqrt{-6})(1-\sqrt{-6})\in I.$$ Now use that $2\in I$ also.