What Defines the Multiplication Rules for Generalized Gaussian Integers?

[gotmilk04]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

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

 

[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?

 

[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?

 

[Dick]

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

 

[gotmilk04]

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

 

[Dick]

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