I know the cyclotomic polynomials can be defined as such (from Wikipedia):
The monic polynomials with integer coefficients that are the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity.
And that they can be described and generated algorithmically.
But what if we take away the "integer coefficient" restriction? I know polynomials like $(x+i)(x+1)(x-1)$ have roots of unity as their roots. So, I have three questions:
If we allow real coefficients, do we get any more polynomials with roots of unity roots?
Is there a general pattern behind or way of generating all complex polynomials with roots of unity as roots?