This seems obvious, but I just can't crack it.
Let K be a field and $F(X) \in K[X]$ be a polynomial. Does $F(a)=0$ for some $a\in K$ imply that F(X) is reducible. Clearly, by the fundamental theorem of algebra, $F(X) = (X-a)G(X)$ for some $G(X) \in \bar{K}[X]$, but does it follow that actually $G(X) \in K[X]$?