Cyclic polynomials and the multiplication matrices of their roots

Abstract

Let D be an integrally closed, characteristic zero domain, K its field of fractions, m⩾2 an integer and P(x)=∑ k=0 m c kx k=∏ i=0 m−1(x−θ i)∈ D[x] a cyclic polynomial. Let τ be a generator of Gal( K[θ 0]/ K) and suppose the θ i are labeled so that τ( θ i )= θ i+1 (indices mod m). Suppose that the discriminant discr K[θ 0]/ K (θ 0,θ 1,…,θ m−1) is nonzero. For 0⩽ i, j⩽ m−1, define the elements a i,j∈ K by θ 0θ i=∑ j=0 m−1 a i,jθ j . Let A=[ a i, j ] 0⩽ i, j< m . We call A the multiplication matrix of the θ i . We have that P( x) is the characteristic polynomial of A. In this article we study the relations between P( x) and A. We show how to factor P( x) in the field K[A] and how to construct A in terms of the coefficients c i . We give methods to construct matrices A, with entries in K , such that the characteristic polynomial of A belongs to D[x] , is cyclic and has A as the multiplication matrix of its roots. One of these methods derives from a natural composition of multiplication matrices. The other method gives matrices A that are generalizations of matrices of cyclotomic numbers of order m, whose characteristic polynomials have roots that are generalizations of (real and complex) Gaussian periods of degree m. As applications we construct families of polynomials with cyclic and dihedral Galois group over Q and with cyclic Galois groups over quadratic fields.