Congruence in Quadratic Integers

[Frillth]

1. The problem statement, all variables and given/known data

For my number theory course, I'm supposed to come up with a definition of congruence in quadratic integers, and define the operations of addition, subtraction, and multiplication.

2. Relevant equations

None known.

3. The attempt at a solution

I honestly have no real idea how to even start this question. Could I please get some help?

[matt grime]

What makes you think it is any different from congruence in the integers?

[Frillth]

I just don't quite grasp the concept. We've only been working with quadratic integers for about two days, and I'm just confused. For example, if we are doing mod i, would every quadratic integer be equal to 0 mod i?

[matt grime]

Does i divide every quadratic integer? Yes: i is a unit un Z[i], that is it has a multiplicative inverse.

It is precisely the same as ordinary modulo arithmetic, and has nothing to do with the quadratic integers per se.