I'm trying to find the minimal polynomial of the 12th primitive root of unity over
Is there a best way of going about this?
I would start by finding the primitive roots then constructing the minimal polynomial from that.
The twelfth roots of unity, of course, satisfy the equation . They are of the form with n from 0 to 11: 1, , , , , , , , , , , and .
A primitive root is one that does NOT satisfy for any smaller value of n which removes 1, -1, i, and -i. What about cube roots?
It might help to note that is equivalent to
Perhaps my terminology isn't quite right, by the "12th primitive root of unity" I meant:
So I am trying to find
As is a root of we have that the min poly divides .
I think perhaps the factorisation is off by power 2?
Hence, the min polynomial is one of the above bracketed terms:
Hence is the min poly.