# 5th Root of Unity

Apr 7 2016

## Calculating the 5th Root of Unity

On the 5th root of unity, the following equation is known.

`cos((2pi)/5) = 1/4 (-1 + sqrt 5)`
We calucuate this value in the following code in 2 ways.

## Naive Approach

`x^4 + x^3 + x^2 + x + 1 = 0`We can divide the above equation with `x^2` (`x != 0` is obvious). Then we get

`(x + 1/x)^2 + (x + 1/x) - 1 = 0`.We can substitute `(x + 1/x)` in this equation with `t`.

`t = x + 1/x`In the above way, we can get the value of the 5th root of unity by solving quadratic equations twice.

## More General Solution Applying Galois Theory

`x^4 + x^3 + x^2 + x + 1 = 0`
The Galois group of this equation has same structure with the multiplicative group of `ZZ//5ZZ-{0}`.

We can utilize this fact to solve this equation as follow.