1. **State the problem:** We want to factorize the expression $$\omega^6 + \omega^5 + \omega^4 + \omega^3 + \omega^2 + \omega + 1$$ where $$\omega$$ is a 7th root of unity, $$\omega \neq 1$$. 2. **Recall the roots of unity:** The 7th roots of unity satisfy $$z^7 - 1 = 0$$ and are given by $$\omega^k = e^{2\pi i k/7}$$ for $$k=0,1,\ldots,6$$. The root $$\omega^0 = 1$$ is excluded. 3. **Use the factorization formula:** We know $$ z^7 - 1 = (z - 1)(z^6 + z^5 + z^4 + z^3 + z^2 + z + 1) $$ so $$ z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 = \frac{z^7 - 1}{z - 1} = 0 \quad \text{for} \quad z \neq 1 \text{ and } z^7=1. $$ 4. **Express as product of linear factors:** The polynomial $$ z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 $$ can be factorized into linear factors corresponding to the other 6 roots of unity: $$ z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 = \prod_{k=1}^6 (z - \omega^k). $$ 5. **Apply to $$\omega$$:** Since $$\omega$$ is one of these roots (with $$\omega \neq 1$$), the expression factorizes as $$ \omega^6 + \omega^5 + \omega^4 + \omega^3 + \omega^2 + \omega + 1 = 0. $$ 6. **Explicit complex factorization:** The factorization into complex linear factors is $$ \boxed{\omega^6 + \omega^5 + \omega^4 + \omega^3 + \omega^2 + \omega + 1 = \prod_{k=1}^6 (\omega - e^{2\pi i k/7})}. $$ This shows the expression is zero exactly at the 7th roots of unity except 1.