Algebraic numbers form a field and they are closed under conjugation, therefore $U=(\alpha+\bar\alpha)/2$ and $V=(\alpha-\bar\alpha)/(2i)$ are algebraic.
Algebraic numbers form a field and they are closed under conjugation, therefore $U=(\alpha+\bar\alpha)/2$ and $V=(\alpha-\bar\alpha)/(2i)$ are algebraic.