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

.