Question

A polynomial, solvable in radicals, whose splitting field is not a radical extension (of $\bf Q$). Let $f(x)$ be any cyclic cubic, that is, any cubic with rational coefficients, irreducible over the rationals, with Galois group cyclic of order 3. Then $f(x)=0$ is solvable in radicals (every cubic is), so the splitting field $K$ of $f$ over $\bf Q$ is contained in a radical extension of $\bf Q$, but $K$ is not itself a radical extension of $\bf Q$. The degree of $K$ over $\bf Q$ is 3, so for $K$ to be radical over $\bf Q$ it would have to be an extension of $\bf Q$ by the cube root of some element of $\bf Q$, but such extensions are not normal.