I'm currently studying the book "Galois Theory" by "David. A. Cox" in page $103$ he tries to prove the following statement:
Given $f_1 \in F_1[x]$ and isomorphism of fields $\phi: F_1 \rightarrow F_2$ there exists an isomorphism $\bar{\phi}: L_1 \rightarrow L_2$ such that $\bar{\phi}{\big|}_{F_1}=\phi$
where in the above $L_1$ is a splitting field of $f_1$ and $L_2$ is a splitting field of $f_2$ which is a polynomial obtained from $f_1$ by applying $\phi$ to its coefficients.
In the course of the proof he uses induction and for the inductive step he writes:
let $f_1(x)=c(x- \alpha_1) \dots (x - \alpha_n)$ then $L_1$ is a splitting of $\frac{f}{x- \alpha_1}$ over $F_1[\alpha_1]$ then if the irreducible factor of $f_1$ containing the root $\alpha_1$ is $h_1$ then $\phi$ must send $h_1$ to an irreducible factor of $f_2$ such as $h_2$,
after that he chooses any root of $h_2$ ($\beta_1$ is the text) and proves that there exists an isomorphism between $L_1$ and $L_2$.
I understand the steps of the proof, my question is that he actually proved a stronger version of the theorem, namely that not only there exists an isomorphism between $L_1$ and $L_2$ such that restricted to $F_1$ is $\phi$ but this isomorphism sends $\alpha_1$ to $\beta_1$, what I don't understand is that how much stronger this really is and what more can we extract from this proof?
As an example if $f_1= h_1 h_2 \dots h_m$ where $h_i$ are irreducible factors, and $f_2= g_1 g_2 \dots g_m$ such that $\phi(h_i)=g_i$ then can we always have a degree of freedom for each $h_i$ for the isomorphism? namely we are free to choose one root from all of the $h_i$ such as $r_1,r_2,\dots,r_m$ and roots $s_1,s_2,\dots,s_m$ from $g_i$ such that $\bar{\phi}(r_j)=s_j$.
I don't believe that this is the case but most of the things that I guess to be true turn out to be false and I also think that there really is something more to take from this proof.
Any help is appreciated