Monogenic even sextic trinomials and their Galois groups

TL;DR

This paper classifies monogenic sextic trinomials of the form x^6 + A x^{2k} + B over the integers, based on their Galois groups, extending previous work and providing explicit descriptions and conditions for their properties.

Contribution

It provides explicit descriptions of all monogenic sextic trinomials with given Galois groups, extending prior results to the case g(x^2) and completing the classification.

Findings

Explicit descriptions for monogenic sextic trinomials with various Galois groups.

Conditions under which infinitely many such trinomials exist.

Analysis of when these trinomials generate distinct sextic fields.

Abstract

Let $f(x)=x^6+Ax^{2k}+B\in {\mathbb Z}[x]$, with $A\ne 0$ and $k\in \{1,2\}$. We say that $f(x)$ is {\em monogenic} if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\theta^3,\theta^4,\theta^{5}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. For each value of $k$ and 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 determine when these descriptions provide infinitely many such trinomials, and we investigate when these trinomials generate distinct sextic fields. These results extend recent work on monogenic power-compositional sextic trinomials of the form $g(x^3)$ to the situation $g(x^2)$, and thereby complete the characterization, in terms of their Galois groups, of monogenic power-compositional sextic trinomials.