On the monogenicity and Galois groups of $\boldsymbol{x^{2p}+ax^p+b^p}$

TL;DR

This paper characterizes when certain irreducible trinomials of the form x^{2p}+ax^p+b^p are monogenic, based on their Galois groups, extending previous research in algebraic number theory.

Contribution

It provides a new characterization of monogenic trinomials x^{2p}+ax^p+b^p using Galois group analysis, expanding prior work.

Findings

Criteria for monogenicity based on Galois groups

Extension of previous results on polynomial monogenicity

Conditions for irreducibility and basis formation

Abstract

Let $f(x)=x^{2p}+ax^p+b^p$, where $p$ is a prime and $a,b\in {\mathbb Z}$ with $ab\ne 0$. If $f(x)$ is irreducible over ${\mathbb Q}$, we say that $f(x)$ is monogenic if $\{1,\theta,\theta^2,\ldots ,\theta^{2p-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In this article, we give a characterization of the monogenic trinomials $f(x)$ according to their Galois groups. These results extend prior investigations of the authors.