Factor polynomial x^8+1

                                                                                                                                                                                    /                 ___________              ___________                ___________                ___________\ /                 ___________              ___________                ___________                ___________\ /               ___________              ___________                ___________                ___________\ /                 ___________              ___________                ___________                ___________\
                                                                                                                                                                                    |                /       ___              /       ___                /       ___                /       ___ | |                /       ___              /       ___                /       ___                /       ___ | |              /       ___              /       ___                /       ___                /       ___ | |                /       ___              /       ___                /       ___                /       ___ |
/         ___________          ___________\ /           ___________          ___________\ /         ___________          ___________\ /           ___________          ___________\ |        ___    /  1   \/ 2       ___    /  1   \/ 2         ___    /  1   \/ 2         ___    /  1   \/ 2  | |        ___    /  1   \/ 2       ___    /  1   \/ 2         ___    /  1   \/ 2         ___    /  1   \/ 2  | |      ___    /  1   \/ 2       ___    /  1   \/ 2         ___    /  1   \/ 2         ___    /  1   \/ 2  | |        ___    /  1   \/ 2       ___    /  1   \/ 2         ___    /  1   \/ 2         ___    /  1   \/ 2  |
|        /       ___          /       ___ | |          /       ___          /       ___ | |        /       ___          /       ___ | |          /       ___          /       ___ | |      \/ 2 *  /   - + -----    \/ 2 *  /   - - -----    I*\/ 2 *  /   - - -----    I*\/ 2 *  /   - + ----- | |      \/ 2 *  /   - - -----    \/ 2 *  /   - + -----    I*\/ 2 *  /   - - -----    I*\/ 2 *  /   - + ----- | |    \/ 2 *  /   - - -----    \/ 2 *  /   - + -----    I*\/ 2 *  /   - + -----    I*\/ 2 *  /   - - ----- | |      \/ 2 *  /   - - -----    \/ 2 *  /   - + -----    I*\/ 2 *  /   - - -----    I*\/ 2 *  /   - + ----- |
|       /  1   \/ 2          /  1   \/ 2  | |         /  1   \/ 2          /  1   \/ 2  | |       /  1   \/ 2          /  1   \/ 2  | |         /  1   \/ 2          /  1   \/ 2  | |            \/    2     4            \/    2     4              \/    2     4              \/    2     4   | |            \/    2     4            \/    2     4              \/    2     4              \/    2     4   | |          \/    2     4            \/    2     4              \/    2     4              \/    2     4   | |            \/    2     4            \/    2     4              \/    2     4              \/    2     4   |
|x +   /   - - -----  - I*  /   - + ----- |*|x + -   /   - - -----  + I*  /   - + ----- |*|x +   /   - + -----  + I*  /   - - ----- |*|x + -   /   - + -----  - I*  /   - - ----- |*|x + - ---------------------- + ---------------------- - ------------------------ - ------------------------|*|x + - ---------------------- - ---------------------- - ------------------------ + ------------------------|*|x + ---------------------- + ---------------------- - ------------------------ + ------------------------|*|x + - ---------------------- + ---------------------- + ------------------------ + ------------------------|
\    \/    2     4        \/    2     4   / \      \/    2     4        \/    2     4   / \    \/    2     4        \/    2     4   / \      \/    2     4        \/    2     4   / \                2                        2                         2                          2            / \                2                        2                         2                          2            / \              2                        2                         2                          2            / \                2                        2                         2                          2            /

$$\left(x + \left(- \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right) \left(x + \left(\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right) \left(x + \left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right)\right) \left(x + \left(- \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right)\right) \left(x + \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) \left(x + \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(x + \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) \left(x + \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right)$$

(((((((x + sqrt(1/2 - sqrt(2)/4) - i*sqrt(1/2 + sqrt(2)/4))*(x - sqrt(1/2 - sqrt(2)/4) + i*sqrt(1/2 + sqrt(2)/4)))*(x + sqrt(1/2 + sqrt(2)/4) + i*sqrt(1/2 - sqrt(2)/4)))*(x - sqrt(1/2 + sqrt(2)/4) - i*sqrt(1/2 - sqrt(2)/4)))*(x - sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2 + sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 - i*sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 - i*sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2))*(x - sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 - sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2 - i*sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 + i*sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2))*(x + sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 + sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2 - i*sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2 + i*sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2))*(x - sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 + sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2 + i*sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 + i*sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2)