Actually, if $N>2$, it's impossible to find $N$ "cyclic polynomials" as you desire.  That's the same as asking if the cyclic invariants are a polynomial ring, which is impossible by the [Chevalley-Shepard-Todd theorem](http://en.wikipedia.org/wiki/Chevalley–Shephard–Todd_theorem)

I suspect in general, the right thing to look at is the sum of the $z_i$'s and all  monomials in $p_h=\sum_{i=1}^N \zeta^{ih}z_i$ where the indices $h$ add to a multiple of $N$ (here $\zeta$ is a primitive $N$th root of unity).