Monogenic period equations are cyclotomic polynomials

摘要

We study monogeneity in period equations, psi(e)(x), the auxiliary equations introduced by Gauss to solve cyclotomic polynomials by radicals. All monogenic psi(e)(x) of degrees 4 <= e <= 250 are determined for extended intervals of primes p = e f + 1, and found to coincide either with cyclotomic polynomials or with simple de Moivre reduced forms of cyclotomic polynomials. The former case occurs for p = e + 1, and the latter for p = 2e + 1. For e >= 4, we conjecture all monogenic period equations to be cyclotomic polynomials. Totally real period equations are of interest in applications of quadratic discrete-time dynamical systems.