Monogenic Even Octic Polynomials and Their Galois Groups

A monic polynomial f (x) Zx of degree N is called monogenic if f (x) is irreducible over Q and \1, , ², , ^{N-1\} is a basis for the ring of integers of Q (), where f () =0. In a series of recent articles, a complete classification of the Galois groups was given for irreducible polynomials \ F (x): =x⁸+ax⁴+b Q[x, \] while a partial classification was given for irreducible polynomials \ G (x): =x⁸+ax⁶+bx⁴+ax²+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.