Let $I$ be a non empty set and $K$ a field. Let $\phi_i=\{f_i(x)\}_{i \in I}$ be a family of separable polynomials. Now, take $L:K$ be a extension given by $K$ and the roots of all polynomials in $\phi$. Prove that $L:K$ is separable and normal, then conclude that $L:K$ is Galois.
I've proved that $L:K$ is separable and normal, but I don't know how can I conclude that $L:K$ is Galois, cause I don't know if $[L:K]$ is finite. Can you have a tip?