Suppose $L:K$ is a splitting field extension for a monic, separable polynomial $f \in K[t]$ which is irreducible over $K$. Let $\alpha_1, \cdots, \alpha_n$ be the distinct roots of $f$ in $L$. Let $k < n/2$ and choose a subset $\beta_1, \cdots, \beta_k, \delta_1, \cdots, \delta_k \subset \alpha_1, \cdots, \alpha_n$ of the roots (all the elements are distinct.)
I know that the maps in $Gal(L:K)$ permute the roots $\alpha_1, \cdots, \alpha_n$. Can I always find an element $\tau \in Gal(L:K)$ such that $\tau(\beta_1) = \delta_1, \cdots, \tau(\beta_k) = \delta_k$?
If not, how do I know which subsets of $\alpha_1, \cdots, \alpha_n$ can be mapped to which other subsets? (Is there a way of knowing this, in general?)