Characterizing monogenic trinomials ${\mathbf{x}}^{12}+\mathbf{a}{\mathbf{x}}^6+\mathbf{b}$ according to their Galois groups

Abstract

Let $f(x)=x^{12}+a{x}^6+b\in \mathbb{Z}[x]$, with $ab\ne 0$. We say that $f(x)$ is {\em monogenic} if $f(x)$ is irreducible over $\mathbb{Q}$ and $\{1,\theta ,{\theta }^2,\dots ,{\theta }^{11}\}$ 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 give explicit descriptions of all monogenic trinomials $f(x)$ having Galois group $G$. These results extend recent work on monogenic power-compositional quartic and sextic trinomials.