Congruence in Quadratic Integers

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Homework Help Overview

The discussion revolves around defining congruence in quadratic integers as part of a number theory course. The original poster expresses confusion regarding the concept and its operations, specifically in relation to modular arithmetic.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • Participants explore the similarities and differences between congruence in quadratic integers and integers. Questions arise about the implications of working with mod i and whether all quadratic integers would be equivalent to 0 mod i.

Discussion Status

Some participants provide insights into the nature of units in Z[i] and suggest that the principles of modulo arithmetic apply similarly to quadratic integers. However, there is no explicit consensus on the original poster's understanding or the definition of congruence itself.

Contextual Notes

The original poster mentions a limited exposure to quadratic integers, having only studied them for a short period, which may contribute to their confusion.

Frillth
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Homework Statement



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.

Homework Equations



None known.

The Attempt at a Solution



I honestly have no real idea how to even start this question. Could I please get some help?
 
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What makes you think it is any different from congruence in the integers?
 
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?
 
Does i divide every quadratic integer? Yes: i is a unit un Z, 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.
 

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