How to check if permutation induces an element of the Galois group.

Question

Let $f \in \mathbb Q[X]$ be irreducible of degree $n$ with zeros $\alpha_1,\dots,\alpha_n \in \mathbb C$. Further, let $L$ be the splitting field of $f$ and $\sigma \in S_n$.

Is there an easy way to check if $ \alpha_i \mapsto \alpha_{\sigma(i)} $ induces an automorphism of $L$?