I know how to prove Euclidean domain for $\mathbb{Z}[\sqrt{2}]$ and $\mathbb{Z}[\sqrt{-2}]$, etc, but I am getting stuck on this notation. From what I've read,
We let $R_d$ be the ring $\mathbb{Z}[(1+\sqrt{2})/2]$ defined as $R_d=\{x + yw : x ,y ∈ \mathbb{Z}\}$, where $w = \sqrt{d}$ if $d$ is not equivalent to $1$ mod $4$, and $(1+\sqrt{d})/2$ if $d$ is equivalent to $1$ mod $4$. So clearly, in this problem, we are assuming that $d= 2$ is equiv. to $1$ mod $4$. But, I really don't know what to do with this information. Does the norm change? What is the norm? Is it the same as it would be for $\sqrt{2}$? I just need a little more information to get going, and I would really appreciate any help you're willing to provide. Thank you.