The monogenicity and Galois groups of certain reciprocal quintinomials

TL;DR

This paper investigates the monogenicity and Galois groups of a specific family of reciprocal quintinomials, extending previous results to new parameter cases and identifying Galois groups in certain instances.

Contribution

It extends prior work on monogenicity of reciprocal quintinomials to include the case where A and B are congruent to 1 mod 4, and determines Galois groups in special cases.

Findings

Extended monogenicity results to A≡B≡1 mod 4 case.

Identified Galois groups for specific parameter choices.

Provided new insights into the algebraic structure of these polynomials.

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 $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ 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 $A\equiv B\equiv 1\pmod{4}$. Moreover, we determine the Galois group over ${\mathbb Q}$ of ${\mathcal F}_{n,A,B}(x)$ in special cases.