How to check if a permutation of roots is a field automorphism?

Question

Let $L$ be the splitting field of a degree $n$ irreducible monic polynomial with integer coefficients over $\mathbb Q$. Let the $n$ roots $\{r_i\}$, then $L = \mathbb Q(r_1,\dots, r_n)$.

Suppose I have computed the roots to a high degree of accuracy with a computer. How could I test if a permutation of the roots was a valid field automorphism?