Let $k_s|K|k$ be a tower of field extensions and $K|k$ be finite and separable ($k_s$ is a separable closure of $k$). There exists $\alpha\in K$ such that $K=k[\alpha]$. Then the splitting field $L$ of the minimal polynomial of $\alpha$ contains $K$ and $L|k$ is a finite Galois extension.
Now I think that the following should be true:
If $\sigma_1,\sigma_2\in\mathrm{Gal}(k_s|k)$ and $\sigma_1|_K=\sigma_2|_K$ then $\sigma_1|_L=\sigma_2|_L$.