Monogenic sextic trinomials $x^6+Ax^3+B$ and their Galois groups
Tom 223 / 2026
Acta Arithmetica 223 (2026), 167-184
MSC: Primary 11R09; Secondary 11R04, 11R32, 11R21
DOI: 10.4064/aa250722-26-8
Opublikowany online: 1 April 2026
Streszczenie
Let $f(x)=x^6+Ax^3+B\in \mathbb Z[x]$, with $A\ne 0$, and suppose that $f(x)$ is irreducible over $\mathbb Q$. We define $f(x)$ to be monogenic if $\{1,\theta ,\theta ^2,\theta ^3,\theta ^4,\theta ^{5}\}$ is a basis for the ring of integers of $\mathbb Q(\theta )$, where $f(\theta )=0$.
For each 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 investigate when these trinomials generate distinct sextic fields.
