Monogenic sextic trinomials $x^6+A{x}^3+B$ and their Galois groups

Abstract

Let $f(x)=x^6+A{x}^3+B\in \mathbb{Z}[x]$, 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.