Find the number of distinct elements.

Question

Let $\omega$ denote a non-real cube root of unity. Then find the number of distinct elements in the set $\{ (1+\omega + \omega^2 + \cdots + \omega^n)^m | m,n \in \Bbb Z_+ \}$

Answer 1

See the $1+\omega+\omega^2+\ldots+\omega^n$ part cycles $\mod 3$. So it suffices to consider $n=0,1,2,3$. For larger values we get one of these four values. Now this gives the part under parentheses is $\{1, -\omega^2, 0, \omega\}$. Now see $m$ cycles $\mod 3$. Thus check for $n=1, 2, 3$. So the set of values is $\{1, -\omega^2,\omega, -1, 0, \omega, \omega^2\}$. I think this is all of it.