9th Root of Unity
Apr 12 2016
Calculating the 9th Root of Unity
`x^9 - 1 = 0`This equation is factorized as follow.
`(x^3 - 1)(x^6 + x^3 + 1) = 0`Then, we solve the following equation.
`x^6 + x^3 + 1 = 0`
The Galois group of this equation has same structure with the multiplicative group of `ZZ//7ZZ-{0}`.
We can utilize this fact to solve this equation as follow.
According to our calculation,
`cos((2pi)/9) = 1/2 (root 3 w + root 3 (w^2))`.