The question is asking you to generalize the idea of congruence in . That is, we write the congruence relation if divides . Can you come up with a similar relation for ?
If α is a quadratic integer in Q[√−d], then define a notion of congruence (mod α). Furthermore, define +, −, and × for congruence classes, and show that this notion is well-defined.
It's not that I don't know how to do this, I just don't understand what the question is asking. Can anyone rephrase this in simpler terms?
Can we just define it in the same way? Let's define a, b, and c such that they are quadratic integers in the given field. If we have the proposition that a = b mod c, then how about we define that as c | (b - a), where the division results in a quadratic integer contained within the given ring of integers. That sounds pretty basic, am I overlooking something?