For several values of $m$, we show ways to construct some families of cyclic monic
polynomials of degree $m$ with integer coefficients and constant terms $\pm 1$, and to
express their roots in terms of Gaussian periods. We give several examples illustrating
those techniques. With the aim to find other methods to construct such families of
polynomials, we consider the question whether one of them can be obtained by means of
rational transformations of a given ordinary family $F(t,X)\in\Z[t][X]$ (at the parameter
$t$) of cyclic monic polynomials of degree $m$ (such families $F$ are easy to find). We
show a method, due to René Schoof, that allows us to answer at least the
simpler question whether a family $G(t,X)\in\Q[t][X]$ of cyclic monic polynomials of
degree $m$ prime with constant term in $\Q^\times$ (independent of $t$ can be obtained
from $F$} by means of rational transformations.