galois theory - Is there a quick way to compute the degree of the splitting field of $x^3+x+1$ over $mathbb{Q}$?

Is there a way to find the degree of the splitting field of $x^3+x+1$ over $\mathbb{Q}$?

Just analyzing the roots shows that the polynomial is separable, so I suppose the splitting field would be a Galois extension. However, the roots are not easy to get a handle on, so it's not obvious to me what roots would need to be adjoined to $\mathbb{Q}$ to get the degree.

What is the right way to do this? Thanks.

Answer

The degree is at least 3 and at most $6=3!$. So you only have to decide whether it's 3 or 6. Since $x^3+x+1$ has only one real root, the other two roots are complex conjugates and so conjugation is an automorphism of the splitting field. Since conjugation has order 2, the degree is 6.

You can avoid Galois theory. The complex roots are roots of a quadratic. Since they cannot be in the real field generated by the real root, the splitting field must be a quadratic extension of the real field and so has degree $2\cdot3=6$.