# Thread: Rewording of a congruence class problem

1. ## Rewording of a congruence class problem

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?

2. The question is asking you to generalize the idea of congruence in $\displaystyle \mathbb{Z}$. That is, we write the congruence relation $\displaystyle a\equiv b\pmod{n}$ if $\displaystyle n$ divides $\displaystyle a-b$. Can you come up with a similar relation for $\displaystyle \mathbb{Q}[\sqrt{d}]$?

3. 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?