Suppose there are more than one polynomials whose coefficients are not all algebraic. Could their product give a polynomial whose coefficients are only algebraic?
I'm asking with regard to factoring over $\overline{\mathbb{Q}}$ univariate or multivariate polynomials over $\overline{\mathbb{Q}}$ by factoring over $\mathbb{C}$.
I already know that values of algebraic functions of more than one non-algebraic numbers could be algebraic.
Say you have $p(x) \cdot q(x)$, monic with coefficients algebraic, and $p$ and $q$ monic. Then the factors have algebraic zeros (subsets of the zeros of the product); as their coefficients are polynomials in their zeros, they are algebraic.