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Fix an embedding of $\overline{\mathbb{Q}}$ into $\mathbb{C}$. Suppose you have a polynomial in several variables, with algebraic coefficients: $P\in \overline{\mathbb{Q}}[z_1, \ldots, z_n]$. Also suppose we find some factorization $P=QR$ for some $Q, R\in \mathbb{C}[z_1, \ldots, z_n]$ (with, a priori, just complex coefficients).

It seems intuitively clear to me that $Q$ and $R$ would need to have algebraic coefficients as well. (Well, almost -- we might need to do some scaling first. Let's say we pick some $\alpha\in\mathbb{C}^\times$ such that $\alpha Q$ has at least one nonzero, algebraic coefficient. Then replace $Q$ and $R$ with $\alpha Q$ and $\frac{1}{\alpha} R$.) Can anyone remind me how you would go about proving this?

(I'm sure this has nothing to do with the particulars of $\overline{\mathbb{Q}}$ and $\mathbb{C}$, really -- we could probably replace them with any pair of nested, algebraically closed fields.)

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