Monogenic even sextic trinomials and their Galois groups

Abstract

Let $f(x)=x^6+A{x}^{2k}+B\in \mathbb{Z}[x]$, 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.