Sufficient condition for reducibility of polynomial $f(x,y)$

Question

[Dual to this question]

Suppose we have a non-constant polynomial $f(x,y)\in\mathbb{Q}[x,y]$ such that the following two conditions are satisfied:

1) For every $x_0\in\mathbb{Q}$, the polynomial $f(x_0, y)\in\mathbb{Q}[y]$ is reducible.

2) For every $y_0\in\mathbb{Q}$, the polynomial $f(x,y_0)\in\mathbb{Q}[x]$ is reducible.

My question is this:

Can we conclude from these two conditions that $f(x,y)$ is a reducible polynomial in $\mathbb{Q}[x,y]$?

Thanks!

Answer 1

Yes. Actually, only one of the conditions needs to hold. This follows from Hilbert's irreducibility theorem.