TITLE: Cyclic cubic numbers as rational linear combinations of roots of unity
QUESTION [8 upvotes]: 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 $\sqrt p$ (where $p$ is an odd prime) 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 ?

REPLY [4 votes]: For the simplest cubic $x^3-ax^2-(a+3)x-1$ which is cyclic and real
with discriminant $p^2$ where $p=a^2+3a+9$, and when $p$ is prime,
the roots $\theta_j,j=0,1,2$ are translates of the Gauss's cubic
periods $\eta_j$. Explicitly $\theta_j=\eta_j+(L-1)/6$, where
$4p=L^2+27$ and $\eta_j$ are the Gauss's periods. Since
$\sum_{j=0}^2 \eta_j=-1$, one can certainly replace $(L-1)/6$ by
linear combinations of $\eta_j$. This relation appeared (and also
the quartic and sextic case) in E. Lehmer, "Connections between
Gaussian periods and cyclotomic units", Maths Comp. Vol 50, No. 182
(1988) 535-541.