Monogenic period equations are cyclotomic polynomials

Abstract

We study monogeneity in {\sl period equations}, ${\psi }_e(x)$, the auxiliary equations introduced by Gauss to solve cyclotomic polynomials by radicals. All monogenic ${\psi }_e(x)$ of degrees $4\le e\le 250$ are determined for extended intervals of primes $p=ef+1$, 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 $e\ge 4$, 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.