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

Abstract

Let $f(x)=x^{12}+ax^{6}+b\in {\mathbb Z}[x]$, with $ab\ne 0$. We say that $f(x)$ is {\em monogenic} if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,\ldots,θ^{11}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. 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.