# Generalized Gaussian integers

1. Sep 13, 2009

### gotmilk04

1. The problem statement, all variables and given/known data
If $$\omega$$ is and nth root of unity, define Z[$$\omega$$], the set of generalized Gaussian integers to be the set of all complex numbers of the form
m$$_{0}$$+m$$_{1}\omega$$+m$$_{2}\omega^{2}$$+...+m$$_{n-1}\omega^{n-1}$$
where n and m$$_{i}$$ are integers.
Prove that the products of generalized Gaussian integers are generalized Gaussian integers.

2. Relevant equations

3. The attempt at a solution
I'm not sure how to start this, so a hint or two would be greatly appreciated.

2. Sep 13, 2009

### Dick

As a first step, you want to prove that w^a*w^b is also an nth root of unity for any integers a and b. Can you handle that?

3. Sep 13, 2009

### gotmilk04

w^a*w^b= w^(a+b)
but what exactly do I need to show to prove it is an nth root of unity?

4. Sep 13, 2009

### Dick

You need to show that the nth power of w^(a+b) is one. That's what a nth root of unity is.

5. Sep 13, 2009

### gotmilk04

Okay, I got that now, but am unsure of the next step and how it relates to the generalized Gaussian integers.

6. Sep 13, 2009

### Dick

Take the product of two of those generalized Gaussian integers. Distribute the product. What kind of terms do you get?

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