Properties that characterize Gaussian periods and cyclotomic numbers

Let $q=ef+1$ be a prime number, $\zeta _q$ a $q$-th primitive root of 1 and $\eta _0,\dots ,\eta _{e-1}$ the periods of degree $e$ of $\mathbb{Q}(\zeta _q)$. Write $\eta _0\eta _i=\sum _{j=0}^{e-1} a_{i,j}\eta _j$ with $a_{i,j}\in \mathbb{Z}$. Several characterizations of the numbers $\eta _i$ and $a_{i,j}$ (or, equivalently, of the cyclotomic numbers $(i,j)$ of order $e$) are given in terms of systems of equations they satisfy and a condition on the linear independence, over $\mathbb{Q}$, of the $\eta _i$ or on the irreducibility, over $\mathbb{Q}$, of the characteristic polynomial of the matrix $[a_{i,j}]_{0\leq i,j\leq e-1}$.