Why do we have $Q(\alpha, z) = \mathbb{Q}(\alpha, z, \overline{z})$ where $\alpha, z, \overline{z}$ are the roots of $X^3+X+1 \in \mathbb{Q}[x]$?

Question

We are given the polynomial $f = X^3+X+1 \in \mathbb{Q}[x]$. It is easy to show that $f$ has only one real root, call it $\alpha$, and the other two roots are complex conjucates: $z, \overline{z}$. Let $L=\mathbb{Q}(\alpha, z, \overline{z})$ be the splitting field. I want to prove that $L = \mathbb{Q}(\alpha, z)$.
Is there a straightforward way to show this? I have a proof for this fact, it is not long, but I feel like there should be a simpler explanation, and maybe a more general one. How would you prove this fact?

Answer 1

$(X-\alpha)(X-z)(X-\overline{z})=X^3+X+1$ so $\alpha+z+\overline{z}=0$, so that $\overline{z}=-z-\alpha$.