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.