Say $f$ is an irreducible polynomial with coefficients in a field $F$. Say $f$ is no longer irreducible over some extension $K$ of $f$, i.e. $f$ factors into a product of (irreducible) polynomials having degree $\geq 0$. Does the splitting field $E$ of $f$ over $F$ necessarily contain $K$?
The splitting field contains every field extension that is generated by adjoining roots of $f$.
It does, of course, not contain every field extension over which $f$ factors. To see this for example take a proper field extension of the splitting field. Clearly $f$ factors there while it is not contained in the splitting field.