Let $L/K$ be a field extension with $K \subset Z_1,Z_2 \subset L$. And furthermore let $M$ be a set completely disjoint with $K,Z_1,Z_2, L$.
If we have $\phi \in Gal(Z_2(M)/Z_1(M))$, then $\phi|_{Z_2} \in Gal(Z_2/Z_1)$. Now if we have $\psi \in Gal(Z_2/Z_1)$, can we extend $\psi$ in such a way that its now in $Gal(Z_2(M)/Z_1(M))$? (by mapping all of $M$ to itself)
And in this way we have a bijective mapping between $Gal(Z_2/Z_1)$ and $Gal(Z_2(M)/Z_1(M))$?
I know the question might be very trivial, but I still can't shake off the feeling that I've made some mistake, thank you.
And to add, is it true that $Gal(Z_2(M)/Z_1(M))$ is Galois if and only if $Gal(Z_2/Z_1)$ is Galois?