Show the Galois group of a polynomial over $\mathbb{Q}$

Question

I need to show what is the Galois group of $x^3-x-1$ over $\mathbb{Q}$

I think I need to first consider whether the polynomial is irreducible over $\mathbb{Q}$, but don't know how to follow from that

Answer 1

The Galois group is isomorphic to a subgroup of $S_3$. The polynomial is irreducible (degree $3$ and has no rational roots), hence the Galois group must be transitive. So it is either $A_3$ or $S_3$.

Now there is a well known theorem which says that the Galois group of an irreducible separable polynomial of degree $n$ is a subgroup of $A_n$ if and only if the discriminant of this polynomial has a square root in the field. Also, the discriminant of a polynomial of the type $x^3+px+q$ is $-4p^3-27q^2$. Can you finish from here?