# 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}.

x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 = 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.