Is there a characterization of irreducible polynomials over $\mathbb Q$ whose splitting field over $\mathbb Q$ are isomorphic to a rupture field?
In other words, of polynomials $P \in \mathbb Q(X)$ that are irreducible over $\mathbb Q$ and that split completely in $\mathbb Q(X) /(P)$.
Equivalently, if $\alpha$ is any root of $P$ then $\mathbb Q(\alpha)$ contains every root of $P$.
Here are the easy cases:
For quadratic polynomials, every rupture field is a splitting field.
For cubic polynomials, a rupture field is a splitting field iff the discriminant is a square.