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?