An elementary characterization of the Galois group of a doubly even octic polynomial

Let Formula: see text be an irreducible polynomial with rational coefficients and Galois group Formula: see text. We extend previous results to give an elementary classification of Formula: see text, identified as a transitive subgroup of Formula: see text up to conjugacy. We show Formula: see text is one of 12 possibilities and can be determined by considering the squareness of at most 11 rational numbers; each number is an expression involving Formula: see text and Formula: see text. We give several applications of our results.