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

TL;DR

This paper classifies monogenic trinomials of the form x^{12}+ax^{6}+b over the rationals based on their Galois groups, providing explicit descriptions for each case.

Contribution

It extends previous work by explicitly characterizing all monogenic trinomials with Galois group G, for each possible Galois group, in the degree 12 case.

Findings

Explicit descriptions of monogenic trinomials for each Galois group G.

Extension of recent results on monogenic quartic and sextic trinomials.

Classification of monogenic degree 12 trinomials based on 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 $\{1,\theta,\theta^2,\ldots,\theta^{11}\}$ 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.