Monogenic Even Octic Polynomials and Their Galois Groups

A monic polynomial f(x)∈ Z[x] of degree N is called monogenic if $f(x)$ is irreducible over Q and \1,θ,θ²,… ,θN-1\ is a basis for the ring of integers of Q(θ), where f(θ)=0. In a series of recent articles, a complete classification of the Galois groups was given for irreducible polynomials \[{ F}(x):=x^8+ax^4+b∈ { Q}[x],\] while a partial classification was given for irreducible polynomials \[{ G}(x):=x^8+ax^6+bx^4+ax^2+1∈ { Q}[x], a≠ 0.\] In this article, for each Galois group G arising in these classifications, we either construct an infinite family of octic monogenic polynomials F(x) or G(x) having Galois group G, or we prove that only a finite such family exists, or we prove that no such octic monogenic polynomial exists. Here, a ``family" means that no two polynomials in the family generate the same octic field. We also provide a minor contribution to the existing partial classification of the Galois groups of G(x) by giving simple conditions on the coefficients a and b to determine when G(x) is monogenic and the Galois group of G(x) is C₂× D₄ versus C₂² C₂.