Suppose $f(x,y)\in\mathbb{Z}[x,y]$ is an irreducible polynomial defining a plane curve. Say I want to find the intersection of this plane curve with the line defined by g(x,y)=y-ax+b. One way to do this would be to find the resultant $Res(f,g;x)$ to find the $y$-coordinates of the affine intersection points and then plug them into the line to get the corresponding $x$-coordinates.
Question: Are there conditions on $f$ and $g$ that guarantee that $Res(f,g;x)$ is an irreducible polynomial in $\mathbb{Z}[y]$? That is, when is it true that the $y$-coordinates in the intersection points are all algebraic conjugates.
I would also appreciate any references.