Suppose $F'/F$ is a field extension. And suppose $f \in F[x]$ is a polynomial. Then $f \in F'[x]$ of course.
So now we may consider two splitting fields for $f$: $K$ over $F$, and $K'$ over $F'$. How does $Gal(K'/F')$ relate to $Gal(K/F)$.
In particular, can we embed $Gal(K'/F')$ into $Gal(K/F)$? How does one construct the homomorphism?