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

Abstract

Let $f(x)=x^{2p}+a{x}^p+b^p$, 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 $\{1,\theta ,{\theta }^2,\dots ,{\theta }^{2p-1}\}$ 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.