Cyclic polynomials and the multiplication matrices of their roots

Abstract

Let D be an integrally closed, characteristic zero domain, K its ÿeld of fractions, m ¿ 2 an integer and P(a cyclic polynomial. Let be a generator of Gal(K[Â0]=K) and suppose the Âi are labeled so that (Âi) = Âi+1 (indices mod m). Suppose that the discriminant discr K[Â 0 ]=K (Â0; Â1; : : : ; Âm−1) is nonzero. For 0 6 i; j 6 m−1, deÿne the elements ai;j ∈ K by Â0Âi = m−1 j=0 ai;jÂj. Let A = [ai;j] 06i; j¡m . We call A the multiplication matrix of the Âi. 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 ÿeld K[A] and how to construct A in terms of the coe cients ci. 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 ÿelds.

Davenport and Hasse's theorems and lifts of multiplication matrices of Gaussian periods

“…, θ d,e−1 } where θ d,i = e−1 s=0 θ s θ ′ ds+i . Thaine [Tha04, Proposition 6] (see also [Tha04,Example 4]) proved that the multiplication matrix A d of θ d,0 , . .…”

“…F. Thaine investigated various properties and characterizations of Gaussian periods, cyclotomic numbers and Jacobi sums with applications to the construction of cyclic polynomials in the series of the papers [Tha96], [Tha99], [Tha00], [Tha01], [Tha04], [Tha08] (see also Lehmer [Leh88], Schoof and Washington [SW88], Hashimoto and Hoshi [HH05a], [HH05b]).…”

“…We organize this paper as follows. In Section 2, we review Thaine's results on the dcomposition of the multiplication matrices based on [Tha04]. In Section 3, we study the dcompositions of matrices and functions in the general situations.…”

Families of cyclic polynomials obtained from geometric generalization of Gaussian period relations

“…As this paper was going to print, the authors learned of the recent paper [21] by Thaine which has closely related material. We also refer to our paper [5] which extends the idea of this paper to meta-cyclic groups.…”