Monogenic sextic trinomials $x^6+Ax^3+B$ and their Galois groups

TL;DR

This paper classifies monogenic sextic trinomials of the form $x^6+Ax^3+B$ over the rationals, detailing their Galois groups and conditions for generating distinct fields, using explicit descriptions and existing theorems.

Contribution

It provides explicit descriptions of all monogenic sextic trinomials with given Galois groups, extending understanding of their algebraic and number-theoretic properties.

Findings

Explicit characterization of monogenic sextic trinomials for each Galois group

Conditions under which these trinomials generate distinct sextic fields

Application of Jakhar, Khanduja, and Sangwan's theorem to classify monogenic cases

Abstract

Let $f(x)=x^6+Ax^3+B\in {\mathbb Z}[x]$, with $A\ne 0$, and suppose that $f(x)$ is irreducible over ${\mathbb Q}$. We define $f(x)$ to be {\em monogenic} if $\{1,\theta,\theta^2,\theta^3,\theta^4,\theta^{5}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. For each possible Galois group $G$ of $f(x)$ over ${\mathbb Q}$, we use a theorem of Jakhar, Khanduja and Sangwan to give explicit descriptions of all monogenic trinomials $f(x)$ having Galois group $G$. We also investigate when these trinomials generate distinct sextic fields.