I'm having a bit of trouble showing that is a Euclidean domain.
No wonder: it is not. For example, check ...
My attempt is:
For this ring, define the norm . We need to show that there is a division algorithm, so let and . We will compute as complex numbers and fish for an appropriate quotient and remainder. We have
Now choose a quotient such that and . Then we have , where is the leftover "fractional" term. From the choice of , we have , and .
At this point, I can't shake the possible equality. I would appreciate any suggestions on this issue.