Galois groups of certain even octic polynomials

Let f(x) = x8 + ax4 + b â š[x] be an irreducible polynomial where b is a square. We give a method that completely describes the factorization patterns of a linear resolvent of f(x) using simple arithmetic conditions on a and b. As a result, we determine the exact six possible Galois groups of f(x) and completely classify all of them. As an application, we characterize the Galois groups of irreducible polynomials x8 + ax4 + 1 â š[x]. We also use similar methods to obtain analogous results for the Galois groups of irreducible polynomials x8 + ax6 + bx4 + ax2 + 1 â š[x].