All the subfields discussed in the following questions are subfields of $\mathbb{C}$.
Given a finite normal extension $L:K$, there exists a tower of subfields $L = K_n \subseteq ... \subseteq K_0 = K$ such that $K_i = K_{i-1}(x_i)$ for some $x_i \in L$.
Thus $$[K_n:K_0] = [K_n:K_{n-1}]...[K_1:K_0]$$
and if we let $p_i$ be the minimal polynomial of $x_i$ over $K_{i-1}$, and $\partial p_i$ be the degree of such polynomial, we can write
$$[K_n:K_0] = \partial p_n ... \partial p_1$$
Noticing that the $\partial p_i$ zeroes of each $p_i$ are different and that for each zeroes $\alpha, \beta$ of any $p_i$ there is a $K$-automorphism $\tau$ of $L$ such that $\tau (\alpha) = \beta$, it seems natural to wonder if the following conjecture is true:
Given the sets {$\alpha_1, ..., \alpha_n$} and {$\beta_1, ... ,\beta_n$} such that $a_i$ and $b_i$ are zeroes of $p_i$ , there exists a $K$-automorphism $\tau$ of $L$ such that, $\tau(\alpha_i)=\beta_i$ for all $i$.
Many theorems of the book I'm reading seem to hint at the statement above being true, yet it has not been proven in any of the chapters I've read. Such statement would, for instance, imply the existence of $[K_n:K_0]$ different $K$-automorphisms of $L$.
I would really appreciate any help/thoughts.