Monogenic even sextic trinomials and their Galois groups

Lenny Jones

Address: Department of Mathematics, Shippensburg University, Shippensburg, Pennsylvania 17257, USA

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Abstract: Let $f(x)=x^6+Ax^{2k}+B\in {\mathbb{Z}}[x]$, with $A\ne 0$ and $k\in \lbrace 1,2\rbrace $. We say that $f(x)$ is monogenic if $f(x)$ is irreducible over ${\mathbb{Q}}$ and $\lbrace 1,\theta ,\theta ^2,\theta ^3,\theta ^4,\theta ^{5}\rbrace $ is a basis for the ring of integers of ${\mathbb{Q}}(\theta )$, where $f(\theta )=0$. 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. These results extend recent work on monogenic power-compositional sextic trinomials of the form $g(x^3)$ to the situation $g(x^2)$, and thereby complete the characterization, in terms of their Galois groups, of monogenic power-compositional sextic trinomials.

AMSclassification: primary 11R09; secondary 11R32, 11R21.

Keywords: monogenic, even sextic, trinomial, Galois, power-compositional.

DOI: 10.5817/AM2026-1-19