On the Construction of Families of Cyclic Polynomials Whose Roots Are Units

Abstract

For several values of m, we show ways to construct some families of cyclic monic polynomials of degree m with integer coefficients and constant terms ±1, and to express their roots in terms of Gaussian periods. We give several examples illustrating those techniques. With the aim to find other methods to construct such families of polynomials, we consider the question whether one of them can be obtained by means of rational transformations of a given ordinary family F(t, X) ∈ ℤ[t][X] (at the parameter t) of cyclic monic polynomials of degree m (such families F are easy to find). We show a method, due to René Schoof, that allows us to answer at least the simpler question whether a family G(t, X) ∈ ℚ[t][X] of cyclic monic polynomials of degree m prime with constant term in ℚ^× (independent of t) can be obtained from F by means of rational transformations.

Keywords:

• Primary 11R09, 11R18, 11T22
• Secondary 11R20, 12F10
• Cyclic polynomials
• units
• Gaussian periods