A monic polynomial f(x) in Z[x] of degree N is called monogenic if f(x) is irreducible over Q and {1,θ,θ², ... ,θ^N-1} is a basis for the ring of integers of Q(θ), where f(θ)=0. In a series of recent articles, complete classifications of the Galois groups were given for irreducible polynomials

F(x):=x⁸+ax⁴+b in Z[x]

and

G(x):=x⁸+ax⁶+bx⁴+ax²+1 in Z[x],

where a is not zero. In this article, for each Galois group G arising in these classifications, we either construct an infinite family of monogenic octic polynomials F(x) or 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.

The author thanks the anonymous referee for a careful reading of this article, and for all the excellent suggestions.