Monogenic even sextic trinomials and their Galois groups

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.