Monogenic Even Octic Polynomials and Their Galois Groups

TL;DR

This paper classifies Galois groups of certain monogenic octic polynomials and constructs infinite families or finite sets of such polynomials for each group, advancing understanding of their algebraic properties.

Contribution

It provides a complete classification of Galois groups for specific monogenic octic polynomials and constructs or enumerates families of polynomials with these groups.

Findings

Constructed infinite families for each Galois group

Identified finite sets of polynomials for certain groups

Enhanced understanding of monogenic octic field structures

Abstract

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In a series of recent articles, complete classifications of the Galois groups were given for irreducible polynomials \[{\mathcal F}(x):=x^8+ax^4+b\in {\mathbb Z}[x]\] and \[{\mathcal G}(x):=x^8+ax^6+bx^4+ax^2+1\in {\mathbb Z}[x], \quad a\ne 0.\] In this article, for each Galois group $G$ arising in these classifications, we either construct an infinite family of monogenic octic polynomials ${\mathcal F}(x)$ or ${\mathcal G}(x)$ having Galois group $G$, or we prove that at most a finite such family exists. In the finite family situations, we determine all such polynomials. Here, a ``family" means that no two polynomials in the family generate isomorphic octic fields.