# 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))`.