Let [Formula: see text] be an irreducible polynomial where [Formula: see text] is a square. We give a method that completely describes the factorization patterns of a linear resolvent of [Formula: see text] using simple arithmetic conditions on [Formula: see text] and [Formula: see text]. As a result, we determine the exact six possible Galois groups of [Formula: see text] and completely classify all of them. As an application, we characterize the Galois groups of irreducible polynomials [Formula: see text]. We also use similar methods to obtain analogous results for the Galois groups of irreducible polynomials [Formula: see text].