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.