It is well known that there are degree 3 irreducible polynomials over $\mathbb Q$ whose splitting fields are not radical. Indeed, there are examples where the splitting field is a real Galois extension of degree 3 and real Galois extensions over $\mathbb Q$ of odd degree cannot be radical.
I was wondering if there are degree 4 irreducible polynomials $f\in \mathbb Q [x]$ whose splitting fields $L$ over $\mathbb{Q}$ are not radical. If the Galois group $G$ of $L$ over $\mathbb Q$ were isomorphic to any transitive subgroup of $S_4$ aside from $A_4$ and $S_4$, then certainly $L$ is radical. This is because these subgroups have a chain of subgroups that are index 2 from each other. The fundamental theorem of Galois theory then allows us to obtain $L$ via consecutive quadratic extensions (and quadratic extensions are radical).
The cases for when $G$ is isomorphic to $S_4$ or $A_4$ remain as possibilities for an example. I was thinking along the lines of finding an irreducible quartic with four real roots having either $S_4$ or $A_4$ as Galois groups, and then adapting the proof of casus irreducibilis. However, I cannot seem to come up with anything.