Monogenic Reciprocal Quartic Polynomials And Their Galois Groups

TL;DR

This paper classifies monogenic reciprocal quartic polynomials with specific Galois groups, identifying all such polynomials over the integers and analyzing their algebraic properties.

Contribution

It explicitly determines all monogenic polynomials of the given form for each possible Galois group, expanding understanding of their algebraic structure.

Findings

Complete classification of monogenic reciprocal quartic polynomials for each Galois group

Explicit descriptions of polynomials with specified Galois groups

Insights into the structure of rings of integers in related number fields

Abstract

Suppose that $f(x)=x^4+Ax^3+Bx^2+Ax+1\in {\mathbb Z}[x]$. We say that $f(x)$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\theta^3\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. For each possible Galois group $G$ that can occur in the two cases of $A\ne 0$ with $B=0$, and $AB\ne 0$, we determine all monogenic polynomials $f(x)$ with Galois group $G$.