Through various examples I have noticed the following result which seems to be true. I do not see how to prove it, nor how to find counterexamples.
Let $P(X)$ be a monic irreducible polynomial with integer coefficients. Let $S$ be any nonempty, proper subset of the set roots of $P$ in a splitting field. Let $\theta$ be an elementary symmetric function in the elements of $S$. Then the discriminant of the minimal polynomial of $\theta$ divides some power of the discriminant of $P$.