Integral bases and monogenity of the simplest sextic fields
Volume 183 / 2018
Acta Arithmetica 183 (2018), 173-183
MSC: Primary 11R04; Secondary 11Y50.
DOI: 10.4064/aa170502-23-10
Published online: 22 March 2018
Abstract
Let $m$ be an integer, $m\neq -8,-3,0,5$ such that $m^2+3m+9$ is square free. Let $\alpha$ be a root of \[ f=x^6-2mx^5-(5m+15)x^4-20x^3+5mx^2+(2m+6)x+1. \] The totally real cyclic fields $K=\mathbb Q(\alpha)$ are called simplest sextic fields and are well known in the literature.
Using a completely new approach we find an explicit integral basis of $K$ in parametric form and we show that the structure of this integral basis is periodic in $m$ with period length 36. We prove that $K$ is not monogenic except for a few values of $m$, in which cases we give all generators of power integral bases.
