Here, $N(\alpha)$ stands for the norm of $\alpha\in\mathbb{G}$, $\mathbb{G}$ is the set of Gaussian Integers, and ($\alpha, \bar\alpha$) is the ideal generated by $\alpha$ and $\bar\alpha$. In other words,
$(\alpha, \bar\alpha) = \{\alpha\lambda + \bar\alpha\mu\ | \ \lambda,\mu\in\mathbb{G}\}$
This problem was asked on a sample exam as a true/false question. I ran into the following problem.
When I attempt to prove ($\alpha, \bar\alpha$) = $\mathbb{G}$, I can show that ($\alpha,\bar\alpha$) $\subset$ $\mathbb{G}$. However, I run into a wall when attempting to show the reverse inclusion. This suggests that the statement is false, but I'm not sure.
Hints would be greatly appreciated!