On the Galois group of a reciprocal even octic polynomial

Let F be a field of characteristic 0, and let f(x)=x8+ax6+bx4+ax2+1∈F[x] be irreducible with Galois group Gal(f). We extend results in the literature to give a complete and elementary classification of Gal(f), identified as a transitive subgroup of S8 up to conjugacy. The approach involves testing whether a small number of elements in F are perfect squares. As an application, we give one-parameter families of polynomials for each of the six possible Galois groups.