Monogenic Even Octic Polynomials and Their Galois Groups

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 $\{1,\theta ,{\theta }^2,\dots ,{\theta }^{N-1}\}$ 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.