# 7th Root of Unity

Apr 7 2016

## Calculating the 7th Root of Unity

On the 7th root of unity, the following equation is known.

`cos((2pi)/7) = 1/6 (-1 + root 3 ((7 + 21 sqrt 3 i) / 2) + root 3 ((7 - 21 sqrt 3 i) / 2))`
We calculate this value in the following code.

We utilize the fact that the Galois group of the following equation has same structure with the multiplicative group of `ZZ//7ZZ-{0}`.

## One more way to solve the equation

`ZZ//7ZZ-{0}` has two normal subgroups whose order are 2 and 3.

The above program uses symmetry derived from the normal subgroup whose order is 2 at first.

The following program uses symmetry derived from the normal subgroup whose order is 3 at first.