Integral bases and monogenity of the simplest sextic fields

TL;DR

This paper provides explicit integral bases for a family of simplest sextic fields, reveals their periodic structure in parameter m, and determines monogenic cases with explicit generators.

Contribution

It introduces a new approach to explicitly compute integral bases of simplest sextic fields and shows their periodicity in parameter m.

Findings

Integral bases are explicitly given in a parametric form.

The structure of the integral basis is periodic with period 36 in m.

The fields are mostly non-monogenic, with specific cases where generators are identified.

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 $K=Q(\alpha)$ are called simplest sextic fields and are well known in the literature. Using a completely new approach we explicitly give an integral basis of $K$ in a 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.