In the book "A Book of Abstract Algebra" by Charles Pinter, page 315, it proved that for a polynomial $a(x)$ over a field $F$, any isomorphism which fixes $F$ sends roots of $a(x)$ to roots of $a(x)$.
The proof is as follows:
Consider $K$ and $K'$ are finite field extensions of $F$, and $K$ and $K'$ have a common extension $E$. If $h:K\rightarrow K'$ is an isomorphism which fixes $F$, and $c \in K$ is a root of a polynomial $a(x) \in F[x]$, then $h(c)$ is also a root of $a(x)$. Hence any isomorphism which fixes $F$ sends roots of $a(x)$ to roots of $a(x)$.
My question is, why is there a need to assume that there exist a common field extension $E$? What is its significance and which part of the proof doesn't work without this assumption?