All complex integers of the same norm = associates?

In summary, the conversation revolves around finding a satisfying definition of "associate" for Gaussian and Eisenstein integers. The definition of an associate being a multiple of the number with one of the roots of unity is considered circular, but it is clarified that this definition applies to Gaussian integers. The conversation also discusses the definition of Gaussian and Eisenstein primes in relation to associates.
  • #1
Ventrella
29
4
Are all complex integers that have the same norm associates of each other?

I have seen definitions saying that an associate of a complex number is a multiple of that number with a unit. And I understand that the conjugate of a complex number is also an associate. But I am looking for a satisfying definition of associate as part of my research on the Gaussian and Eisenstein integers.

I have found a satisfying definition of a Gaussian prime or an Eisenstein prime as being "only divisible by a root of unity and one of its associates". But now I want to find a definition of "associate". The definition I found is "a multiple of the number with one of the roots of unity". This is a circular definition. If the definition of the associate of a Gaussian or Eisenstein integer p were "all integers that have the same norm as p", then that would be great. But since I haven't found this definition, I wonder if I am missing something?

Thanks!
-Jeffrey
 
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  • #2
Ventrella said:
I have found a satisfying definition of a Gaussian prime or an Eisenstein prime as being "only divisible by a root of unity and one of its associates". But now I want to find a definition of "associate". The definition I found is "a multiple of the number with one of the roots of unity". This is a circular definition.
It is not circular. In the Gaussian integers the roots of unity are 1, i, -1 and -i. The associates of a number z are z, iz, -z, -iz. Now you can use this definition to define primes.

Using your definition, 3+4i, 3-4i and 5 are not associates, despite having the same norm.
 

1. What are complex integers?

Complex integers, also known as Gaussian integers, are numbers of the form a + bi, where a and b are integers and i is the imaginary unit (√-1).

2. What is the norm of a complex integer?

The norm of a complex integer is the square of its absolute value, or the product of the complex integer and its complex conjugate. In other words, it is the sum of the squares of the real and imaginary parts of the complex integer.

3. What does it mean for complex integers to have the same norm?

When two complex integers have the same norm, it means that their absolute values are equal. This also means that their distance from the origin on the complex plane is the same.

4. What are associates of a complex integer?

Associates of a complex integer are other complex integers that have the same norm. They are essentially different representations of the same number on the complex plane.

5. Why are all complex integers of the same norm considered associates?

This is because they share the same properties and behave in the same way in mathematical operations. Therefore, they can be considered equivalent or interchangeable in certain contexts.

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