The monogenicity and Galois groups of certain reciprocal quintinomials

Abstract

We say that a monic polynomial $f(x)\in \mathbb{Z}[x]$ of degree $N$ is monogenic if $f(x)$ is irreducible over $\mathbb{Q}$ and $\{1,\theta ,{\theta }^2,\dots ,{\theta }^{N-1}\}$ is a basis for ${\mathbb{Z}}_K$, the ring of integers of $K=\mathbb{Q}(\theta )$, where $f(\theta )=0$. For $n\ge 2$, we define the reciprocal quintinomial \[{\mathcal F}_{n,A,B}(x):=x^{2^n}+Ax^{3\cdot 2^{n-2}}+Bx^{2^{n-1}}+Ax^{2^{n-2}}+1\in {\mathbb Z}[x].\] In this article, we extend our previous work on the monogenicity of ${\mathcal{F}}_{n,A,B}(x)$ to treat the specific previously-unaddressed situation of $A\equiv B\equiv 1(\mathrm{mod}4)$. Moreover, we determine the Galois group over $\mathbb{Q}$ of ${\mathcal{F}}_{n,A,B}(x)$ in special cases.

arXiv admin note: text overlap with arXiv:2404.17921