Can a bivariate polynomial with only algebraic coefficients have monic polynomial factors with non-algebraic coefficients?

Question

Can a bivariate polynomial with only algebraic coefficients have monic polynomial factors with non-algebraic coefficients?

Are there literature references with lists of bivariate polynomials that are irreducible over the algebraic numbers?

I already know that a bivariate polynomial can have solutions $(z_1,z_2)\in\mathbb{C}^2$.

Answer 1

The quick answer is no:

Anytime you factor a polynomial of one or several variables $f \in K[x_1, \ldots, x_n]$ you can rescale the factors so that the coefficient of each factor is algebraic over the field generated by the coefficients of $f$. The reason is that each polynomial can be factored in (essentially) finitely many ways into two factors.