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