TITLE: Is there a way to separate the real and the imaginary parts of the roots of a polynomial without knowing the roots of the polynomial?
QUESTION [6 upvotes]: Given a first polynomial, is it possible to construct a second polynomial whose roots are only the imaginary (or likewise, only the real) parts of the roots of the first polynomial without knowing the roots of the first polynomial?
I think it is not possible, but I am interested in hearing your thoughts on this.

REPLY [2 votes]: You can start with a normalized polynomial of second degree $p(z)=(z-z_1)(z-z_2)$.
If we know the roots of the first polynomial, we can formulate the second polynomial
$$g(z)=\left(z-\dfrac{z_1+\overline{z}_1}{2}\right) \left(z-\dfrac{z_2+\overline{z}_2}{2}\right).$$
The overline designates the complex conjugate.
If you expand this expression you will see that there is not an simple expression depending on $p(z)$.
Hence, I would conclude that it is not possible to do what you asked for. You can try a similar approach for the imaginary part of the roots with $(z-\overline{z})/2$ for the imaginary part of the corresponding root. Only the trivial linear polynomial allows this procedure because we directly know the roots.