Let $f \in \mathbb Q[X]$ and $f(X) = (X - \alpha_1)\cdots (X - \alpha_n)$ with $\alpha_1, \dots, \alpha_n \in \mathbb C$ pairwise distinct. Let $L = \mathbb Q(\alpha_1, \dots, \alpha_n)$ be the splitting field of $f.$
Furthermore, suppose that for every permutation $\pi$ on $\{1, 2, \dots, n\}$, there exists an automorphism $\sigma \in \text{Aut}_{\mathbb Q}(L)$ with $\sigma(\alpha_i) = \alpha_{\pi(i)}$ for every $i = 1, 2, \dots, n.$
I have to show the polynomial $f$ is irreducible over $\mathbb Q$, but I am stuck.
I figured out that $\alpha_i$ cannot be in $\mathbb Q$ for any $i$, but how do I know that there cannot be a factorization of a different degree? Any help would be appreciated!
Here’s a hint: suppose that $f=gh$ with $g,h$ non-constant polynomials with rational coefficients. WLOG, you may assume that $g(\alpha_1)=0$.
Now, what kind of root of $f$ may $\sigma(\alpha_1)$ be, for any $\sigma \in \mathrm{Aut}_L(\mathbb{Q})$? So what can you deduce about $\pi(1)$ for any permutation $\pi$ of $\{1,\ldots,n\}$?