6. Let $\omega=\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}$, then the equation with roots $\omega, \omega^{3}, \omega^{7}, \omega^{9}$ is
(A) $x^{4}+x^{3}+x^{2}+x+1=0$
(B) $x^{4}-x^{3}+x^{2}-x+1=0$
(C) $x^{4}-x^{3}-x^{2}+x+1=0$
(D) $x^{4}+x^{3}+x^{2}-x-1=0$
(B)
6.【Analysis and Solution】This problem can also be solved using the inspection method. Clearly, $|\omega|=1, \omega^{10}=1$, so $\omega+\omega^{3}+\omega^{7}+\omega^{9}=\omega+\omega^{3}+\overline{\omega^{3}}+\bar{\omega}=2 \cos \frac{\pi}{5}+2 \cos \frac{3 \pi}{5}$
$$
=4 \cos \frac{2 \pi}{5} \cos \frac{\pi}{5}=4 \cos \frac{2 \pi}{5} \cos \frac{\pi}{5} \frac{\sin \frac{\pi}{5}}{\sin \frac{\pi}{5}}=1 .
$$

By the relationship between roots and coefficients, we can exclude (A) and (D). Also,
$$
\omega \omega^{3} \omega^{7}+\omega \omega^{3} \omega^{9}+\omega \omega^{7} \omega^{9}+\omega^{3} \omega^{7} \omega^{9}=\omega+\omega^{3}+\omega^{7}+\omega^{9}=1 \text {, }
$$

we can further exclude (C), thus the answer is (B).