  When using a Groebner basis to solve the highly symmetric system of algebraic
equations defining the cyclic p-roots, one has the feeling that much of the
advantage of computerized symbolic algebra over hand calculation is lost
through the fact that the symmetry is immediately ``thrown out'' by the
calculations. In this paper, the problem of finding (for all relevant primes p)
all cyclic p-roots of index 3 is treated with the symmetry preserved through
the calculations. Once we had found the relevant formulas, using MAPLE and
MATHEMATICA, the calculations could even be made by hand. On the other hand,
with respect to a straightforward attack with Groebner basis, it is not even
clear how this could be organized for a general p.
  In other terminologies, our results involve listings of all bi-unimodular
sequences constant on the cosets of the group G_0 of cubic residues, or
equivalently all circulant complex Hadamard matrices related to G_0.
  The corresponding problem for bi-unimodular sequences of index 2 was solved
by the first named author in 1989 and shortly after solved independently by de
la Harpe and Jones in the case p = 1 (mod 4) and by Munemasa and Watatani in
the case p = 3 (mod 4).
