In the written version of a talk Barry Mazur gave to Friends of the Harvard Mathematics Department on May 5, 2009, there is an interesting question in Footnote 5 (page 8).

He recalls how Gauss wrote $\sqrt p$ as an explicit rational linear combination of roots of unity (using Gauss sums) and says that he doesn't know any such explicit expression for the roots $\alpha$ of an irreducible cubic polynomial $T^3+bT+c\in\mathbf{Q}[T]$ whose discriminant is a square (so that $\mathbf{Q}(\alpha)$ is a cyclic extension of $\mathbf{Q}$, and hence contained in $\mathbf{Q}(\zeta)$ for some root of unity $\zeta$).

Question. Does anyone know such an explicit expression for the roots of irreducible cubic polynomials whose discriminant is a square ?