Monogenic even sextic trinomials and their Galois groups
Lenny Jones
TL;DR
The paper fully characterizes monogenic sextic trinomials of the form $f(x)=x^6+Ax^{2k}+B$ with $k\in\{1,2\}$ by linking their monogenicity to the possible Galois groups over $\mathbb{Q}$ through the Jakhar–Khanduja–Sangwan framework. It proves that no such polynomial has Galois group $C_6$ or $S_3$, and provides explicit necessary-and-sufficient conditions for the remaining groups, including $C_2\times S_3$, $A_4$, $C_2\times A_4$, $S_4^{+}$, $S_4^{-}$, and $C_2\times S_4$, with both parametric descriptions and finite/listed instances. The results combine discriminant analysis, local $p$-adic criteria, and elliptic-curve techniques to determine monogenicity and to construct infinite families of pairwise-distinct sextic fields. Corollaries then produce infinite one-parameter subfamilies yielding distinct sextic fields for several Galois types, demonstrating rich arithmetic diversity among monogenic sextic trinomials.
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 monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,θ^3,θ^4,θ^{5}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. 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.