# 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.