Let $C$ be the field of complex numbers. Polynomial $f \in K[x,y]$ is called closed or non-decomposible if $C[f]$ is algebraically closed (definition for $f \in K[x1,...,xn]$ is the same).
Theorem. Next conditions are equivalent: (1) $f$ is closed;
(2) there does not exist any $F \in K[t]$ such that $f = F(h)$ for some $h \in K[x,y]$;
(3) $f + a$ is irreducible for all except finitely many $a \in C$.
(4) there exist $a \in C$ such that $f + a$ is irreducible. (end of the theorem)
It is clear, if $f$ is irreducible then $f$ is closed.
It is easy to see that if $f$ and $g$ are irreducible, then $fg$ is closed.
Question: Assume $f$ and $g$ are closed. Is it true that $fg$ is closed?