The coefficients of the period polynomials

A general description of the Vi`ete coefficients of the gaussian period polynomials is given, in terms of certain symmetric representations of the subgroups and the corresponding quotient groups of the multiplicative group $\mathbf{F}$_{p}^{*} of a finite prime field of characteristics p, an odd prime number. The known values of these coefficients are recovered by this technique and further results of general nature are presented.