Let $K/F$ be a finite field extension. If you have a polynomial $f(x)=\prod_{r\in R} (x-r)$ whose splitting field is $K$ and $r \notin F, \forall r\in R$ (the set of roots), does $\prod_{r\in X} (x-r) \notin F[x]$ for all subsets $X$ of $R$ imply that $f$ is irreducible over $F$ ?
For example: If $f(x) = (x-r_{1})(x-r_{2})(x-r_{3})(x-r_{4}), r_{i} \notin F$. Does $(x-r_{j})(x-r_{k})\notin F[x]$ for any $r_{j},r_{k}$ imply f is irreducible?