Hello everyone,

I'm having a bit of trouble showing that

is a Euclidean domain.

No wonder: it is not. For example, check ...

Tonio
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.