Teil 1

Nullstellen: ´x^6 = -1 = e^(i pi)´

´x^6 = e^(i pi) = e^(i (pi + k * 2pi))´ ´k in ZZ´

´=> x_k = e^(i((pi + k * 2pi)/6))´

´k: 0,1,…,6´ ´=> x^6+1 = (x - e^(i pi/6)) * ´´(x - e^(i (3pi)/6)) * ´´(x - e^(i (5pi)/6)) * ´´(x - e^(i (7pi)/6)) * ´´(x - e^(i (9pi)/6)) * ´´(x - e^(i (11pi)/6))´

Umwandlung in arithmetische Form fehlt!

Teil 2

´(x - e^(i pi/6))(x - e^(i (11pi)/6))´ ´= x^2 - (e^(i pi/6) + e^(i (11pi)/6)) + e^(i pi (1/6 + 11/6))´ ´= x^2 - Re(e^(i pi/6))x + 1´

´x^6 + 1 = (x^2 - cos(20˚)x + 1) * ´´(x^2 - cos(40˚)x + 1) * ´´(x^2 - cos(60˚)x + 1) * ´´(x^2 - cos(80˚)x + 1) * ´´(x^2 - cos(100˚)x + 1) * ´´(x^2 - cos(120˚)x + 1)´

Teil 3

´-1 = 1´ in ´ZZ_2´

´x^6 + 1 = x^6 - 1´ ´= (x^3)^2 - 1^2´ ´= (x^3 + 1)(x^3 - 1)´ ´= (x^3 - 1)^2´ ´= (x^3 - 1)^2´

Ende fehlt

Für ´x^8 + 1´:

´x^8 + 1 = x^8 - 1´ ´= (x^4)^2 - 1^2´ ´= (x^4 + 1)(x^4 - 1)´ ´= (x^4 - 1)^2´ ´= ((x^2)^2 - 1^2)^2´ ´= ((x^2 + 1)(x^2 - 1))^2´ ´= ((x^2 - 1)^2)^2´ ´= (x^2 - 1)^4´ ´= ((x - 1)(x + 1))^4´ ´= (x + 1)^8´