Degree of Minimal polynomial of complex number and its components

Question

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?

Answer 1

$$ \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} = \textrm{exp}\left( \frac{i \pi}{4} \right) $$ is (probably, if I understood the statement correctly) a counterexample.

Answer 2

For any root of unity $\zeta$, of order at least 5, we have that its real part is $\frac12(\zeta +\bar\zeta)$ and its degree would be half of the degree of $\zeta$. Galois theory tells us that this number is invariant under the automorphism sending $\zeta $ to $\zeta^{-1}$ and hence of smaller degree, belonging to a proper subfield.