The product of any two of the tenth roots of | vedclass

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(A) The $n$-th roots of unity are given by $z_k = e^{i \frac{2k\pi}{n}}$ for $k = 0, 1, \dots, n-1$.For $n=10$,the roots are $\omega^k = e^{i \frac{2k

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Always one of the tenth roots of unity

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(A) The $n$-th roots of unity are given by $z_k = e^{i \frac{2k\pi}{n}}$ for $k = 0, 1, \dots, n-1$.For $n=10$,the roots are $\omega^k = e^{i \frac{2k\pi}{10}}$ where $k \in \{0, 1, \dots, 9\}$.Let $z_1 = \omega^r$ and $z_2 = \omega^s$ be two such roots,where $r, s \in \{0, 1, \dots, 9\}$.The product is $z_1 z_2 = \omega^r \cdot \omega^s = \omega^{r+s} = e^{i \frac{2(r+s)\pi}{10}}$.Since $\omega^{10} = 1$,we have $\omega^{r+s} = \omega^{(r+s) \pmod{10}}$.Since $(r+s) \pmod{10}$ is always an integer in the set $\{0, 1, \dots, 9\}$,the product $z_1 z_2$ is always one of the tenth roots of unity.

The product of any two of the tenth roots of unity is

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