Let $p(x)$ be a polynomial of degree $n$ over a field $F$. Let $K/F$ be an extension that contains all the roots $\alpha_1, \ldots , \alpha_n$ of $p(x)$. Suppose that the sum of any the roots of $p(x)$ is not in $F$, unless the sum is 0 or the sum includes all the roots.
To prove: $p$ is irreducible over $F$.
Attempt: Suppose $p$ is reducible. Write $p(x) = g(x)h(x)$ where both $g$ and $h$ have degree $< n$. Consider $g$. Then $g$ has at most $n - 1$ roots, so some $\alpha_i$ is not a root of $g$. Then...?
I really don't see how to use the condition on the sums of the roots and how they relate to the (ir)reducibility of $p$.
Hints?