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.
Reference: Lang's Algebra (Springer 2002), Chapter V, Proposition 1.6
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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. |
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