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? 



product of two non-decomposible (closed) polynomials
As I mentioned above in the comment, the answer to my question (very easy question) is negative!


P.S. Of course I meant that $f$ and $g$ are algebraically independent!


