Degree of splitting field of $f(x) \in \mathbb Q[x]$

Question

I'm having trouble with a homework question, and I'm wondering if I can get some tips and/or hints (I do not want someone to solve the problem for me).

The question is as follows:

Let $f(x) \in \mathbb{Q}[x]$ be an irreducible monic polynomial of degree 3 that does not split over $\mathbb{R}$. Find the degree of the splitting field of $f(x)$ over $\mathbb{Q}$.

I haven't gotten far, but this is my work:

Since $f(x)$ does not split over $\mathbb{R}$, we know that it has a complex root with a non-zero imaginary part. Denote this root by $\alpha$. By the complex conjugate root theorem, $\overline{\alpha}$ is also a root of $f(x)$, that is $f(\overline{\alpha})=0$. I then tried to check if it is true that $\overline{\alpha} \in \mathbb Q(\alpha)$, but I can't seem to reach a conclusion (however, my intuition says that it is not true). I also know that $f(x)$ must have a real root, since $\deg(f(x))=3$ (so it can't have another complex root by the conjugate root theorem).

Any help is appreciated!

Thanks!

Answer 1

Hint: Use this general statement, which can be proved by induction:

The degree of the splitting field of a polynomial of degree $n$ is at most $n!$