The elements of the set $U_n = \{z \in \mathbb{C} : Z^n =1 \}$ are called the $n^{\text{th}}$ roots of unity. Using the technique of Examples 1.6 and 1.7, we see that the elements of this set are the numbers $$\cos \left(m \frac{2\pi}n \right)+i \sin \left(m \frac{2\pi}n \right) \quad \text{for} \quad m=0,1,2,\ldots,n-1$$ They all have absolute value $1$, so $U_n \subset U$. If we let $\zeta = \cos \frac{2\pi}n + i \sin \frac{2\pi}n$, then these $n^{\text{th}}$ roots of unity can be written as $$1=\zeta^0,\zeta^1,\zeta^2,\zeta^3,\ldots,\zeta^{n-1} \tag{10}$$ Because $\zeta^n=1$, these $n$ powers of $\zeta$ are closed under multiplication. For example, with $n=10$, we have $$\zeta^6\zeta^8 = \zeta^{14} = \zeta^{10} \zeta^4 = 1 \cdot \zeta^4 = \zeta^4$$
After some googling I found de Moivre's formula that explains why the roots of unity can be written as they are in equation (10). So far I get it. But I don't understand why the textbook says these roots are equal to 1.
Shouldn't it be like this:? $$ 1 = (\zeta^0)^n , (\zeta^1)^n , (\zeta^2)^n , (\zeta^3)^n , ... , (\zeta^{n-1})^n , $$
Is it an error in my textbook or I have I misunderstood what I am reading here?
