I've been doing some research on the constructibility of regular polygons, and it led me to come up with the following conjecture:
Edit (Sorry I should have imposed much stricter conditions):
Let $z$ be a principal root of unity. Then the degree of its minimal polynomial is equal to the degree of the minimal polynomial of both of its components.
So far I've used de Moivre's Formula and Chebyshev polynomials to show that degree of the minimal polynomial of $z$ divides the degree of the minimal polynomial of its real component.
Can anyone give a complete proof or counter-example?