This irreducible polynomial has 1 real root and 2 complex roots that are complex conjugates. If is complex conjugation then we see that is a transposition (2-cycle) if viewed as an element of (remember that the Galois group is a subgroup of ). Now if is a root of then is a degree extension and so divides the degree of the splitting field over . Thus, divides the order of the Galois group, hence there is an element of order 3 in the Galois group i.e. a 3-cycle. But any subgroup of generated by a 2-cycle and a 3-cycle is itself, so the Galois group is .