Cyclic cubic numbers as rational linear combinations of roots of unity 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 ?
 A: I guess Mazur's remark just means that in the expression of the cubic polynomial whose roots generate the cubic subfield of the $p$-th roots of unity, there are numbers $L$ and $M$ with $L^2 + 27M^2 = p$. It reflects the fact from Kummer theory that for understanding cyclic cubic extensions you have to adjoin the cube roots of unity, whereas for quadratic extensions the necessary square roots of unity are already there.
The computation of the cubic subfield of a cyclotomic field with prime conductor using "Gaussian periods" is already in the Disquisitiones, but you certainly should be able to find a modern exposition in Cohen's books.
A: 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.
