Subset of C2 such that any vanishing cubic polynomial is reducible

Question

I'm looking for a subset of $ \mathbb C^2$ consisting of four points such that any polynomial of degree 3 vanishing on this set is reducible.

Don't really know where to start, any ideas on how to approach this are appreciated.

Answer 1

Take any four points that lie on a line will do. After a change of coordinates suppose your four points are given by $(0,0), (0,1), (0,2), (0,3)$, and let $f(x,y)$ denote your cubic polynomial.

Then by assumption, we have that $f(0,y)$ has four roots $y=0,1,2,3$. But $f(0,y)$ has degree at most $3$, so if it has four roots it must imply that $f(0,y)=0$. But this implies that $f(x,y) = xg(x,y)$ for some $g$.