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

All cyclic p-roots of index 3, found by symmetry-preserving calculations