Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers

Given an odd prime $p$ we show a way to construct large families of polynomials $P_{q}(x)\in \mathbb{Q}[x]$, $q\in \mathcal{C}$, where $\mathcal{C}$ is a set of primes of the form $q\equiv 1$ mod $p$ and $P_{q}(x)$ is the irreducible polynomial of the Gaussian periods of degree $p$ in $\mathbb{Q}(\zeta _{q})$. Examples of these families when $p=7$ are worked in detail. We also show, given an integer $n\geq 2$ and a prime $q\equiv 1$ mod $2n$, how to represent by matrices the Gaussian periods $\eta _{0},\dots ,\eta _{n-1}$ of degree $n$ in $\mathbb{Q}(\zeta _{q})$, and how to calculate in a simple way, with the help of a computer, irreducible polynomials for elements of $\mathbb{Q}(\eta _{0})$.