Let \(D\) be an integrally closed, characteristic zero domain, \(K\) its field of fractions, and let \(P(x)=\sum_{k=0}^m c_kx^k= \prod_{i=0}^{m-1}(x-\theta_i)\in D[x]\) be a cyclic polynomial. Let \(\tau\) be a generator of \(\text{Gal}(K(\theta_0)/K)\), suppose that \(\tau(\theta_i)=\theta_{i+1}\) (indices mod \(m\)), and \(\text{discr}(\theta_0,\dots,\theta_{m-1})\) is nonzero. The multiplication matrix \(A=\begin{pmatrix} a_{ij}\end{pmatrix}\) is defined by \(\theta_0\theta_i= \sum a_{ij}\theta_j\). Then \(P(x)\) is the characteristic polynomial of \(A\). In the first section the author gives a complete characterization of the multiplication matrices, and shows how they change under translations and linear transformations of the roots. It is further shown how to construct \(A\) in terms of the coefficients \(c_i\). Explicit computations are given for \(m\leq4\). In the next two sections, the author develops a general method for how to construct such matrices \(A\). There is an interesting definition of composition of multiplication matrices. In particular, there are generalizations of matrices of cyclotomic numbers with characteristic polynomials whose roots are generalizations of Gaussian periods. It is shown how to construct generalized Jacobi sums by using Stickelberger elements and certain roots of unity. As applications, the author finds families of polynomials with cyclic and dihedral Galois groups over \(\mathbb Q\), and with cyclic Galois groups over quadratic fields.