# 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)).